Method and system for subband encoding and decoding of an overcomplete representation of the data structure

ABSTRACT

A bit stream representing n-dimensional data structures may be encoded and decoded. A part of the data can be mappable within predefined similarity criteria to a part of the data of another data structure. The similarity criteria may include, a spatial or temporal shift of the data. The data structures are typically sequential video frames such as is used in motion estimation and/or compensation of moving pictures, and a part of the data structure may be a block of data within a frame. The shift may be any suitable shift such as linear translation, rotation, or change of size. Digital filtering may be applied to a reference or other frame of data to generate subbands of a set of subbands of an overcomplete representation of the frame by calculations performed at single rate. The digital filtering may be implemented in a separate filter module or in software.

RELATED APPLICATIONS

This application claims priority to, and hereby incorporates byreference in its entirety, provisional U.S. Patent Application No.60/317,429, filed on Sep. 4, 2001, and entitled ‘IN-BAND MOTIONCOMPENSATION WAVELET VIDEO ENCODERS AND DECODERS.’ Further, thisapplication claims priority to, and hereby incorporates by reference inits entirety, provisional U.S. Patent Application No. 60/361,911, filedon Feb. 28, 2002, and entitled ‘IN-BAND MOTION COMPENSATION WAVELETVIDEO ENCODERS AND DECODERS’.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to methods of encoding and decoding a bit streamcomprising a representation of a sequence of n-dimensional datastructures or matrices, in which n is typically 2. The invention isparticularly relevant to in-band motion estimation/motion compensationof video images.

2. Description of the Related Art

Wavelet-based coding has been generally accepted as the most efficienttechnique for still-picture compression. Wavelet transform schemes aredescribed in detail in “Wavelets and Subbands”, by Abbate, DeCusatis andDas, Birkhäuser press, 2002. The insertion of discrete wavelettransforms (DWT) in the new JPEG-2000 coding standard led to increasedcoding efficiency in comparison to previous standards in this area, andadditionally, a number of interesting features including quality andresolution scalability, stemming from the multiresolution nature of thetransform are provided. In the video-coding arena, although suchscalability features (along with temporal scalability) are highlydesired in a number of applications (like video streaming and multimediaover networks), wavelets are employed only for the texture coding in theMPEG-4 standard at present. To address scalability, the MPEG-4 standardadopts the multiresolution DCT approach within a hybrid codingstructure, which performs relatively poorly in the complexity versuscoding efficiency sense in comparison to wavelets. For these reasons,many authors have begun to explore wavelet-based scalable video-codingschemes. Until recently, the research efforts were mainly directedtowards the use of 3-D wavelet decompositions for each input group offrames (GOF) in order to remove the spatial and the temporalredundancies in the video stream. This work was pioneered mainly byKarlsson and Vetterli [1], Lewis and Knowles [2], and more recently byOhm [3] and Taubman and Zakhor [4] who introduced 3-D decompositionscoupled with motion estimation (ME) and motion compensation (MC). Morerecent algorithms proposed by Kim, Xiong and Pearlman [5] and Bottreauet al [6] support all types of scalability (spatial, quality andtemporal by using 3-D versions of the SPIHT algorithm [7] andhierarchical spatial-domain techniques for block-based ME and MC. Awavelet decomposition using a short filter-pair like the Haar transformis performed in the temporal direction to remove the redundanciesbetween successive residual frames. Furthermore, a 2-D waveletdecomposition of the motion compensated sequence (i.e. the residualframes) is performed to reduce spatial redundancies and to compact theenergy in the lower-frequency subbands (using classical filters fromstill-image coding, such as the 9/7 filter-pair). Quality scalabilitycan be obtained with this type of algorithms by coding thethree-dimensional transform-domain coefficients using the 3-D extensions[5] of the classical 2-D embedded zerotree-based [7] or block-basedwavelet image coders [8][9]. Spatial scalability can be achieved only ifthe motion compensation is performed in a level-by-level manner. Inaddition, temporal scalability is inherent to such schemes, since in amultilevel temporal decomposition each resolution reconstructs to adynamically-reduced frame-rate for the decoded sequence. In conclusion,these schemes algorithmically satisfy the scalability issues, andmoreover, they provide good coding performance. Nevertheless, theirlimitation comes from the implementation point of view because theyrequire a large memory budget for the 3-D transform-application to eachGOF, and they distribute almost equally the computational load betweenthe encoder and the decoder, thus making the decoder implementationrelatively complex. In addition, the complete codec delay is alsoincreased since the decoder can receive compressed data only after thefull 3-D transform is completed in the current GOF of the encoder. Thusthey are insufficient for bi-directional communications and forapplications where power dissipation and memory are major cost issues,i.e. for portable systems. Other approaches for scalable wavelet videocoding which try to reduce the implementation complexity and the systemdelay follow the classical MPEG-alike hybrid coding-structure, where theME/MC is also performed in the spatial domain and the DCT transform isreplaced with a wavelet transform. Typical examples of such systems aredescribed in [10] and [11]. Although scalability in quality can beachieved by embedded wavelet coding [7][8][9], the main drawback of suchtechniques is that they fail to take advantage of the inherentmultiresolution structure of the wavelet transform to provide drift-freespatial scalability. In addition, there is an inverse transform in thecoding loop, resulting to two transform-applications (one forward andone inverse) per frame and per spatial resolution. This may also lead tolarge codec delays, since no parallelism is possible and each wavelettransform is applied to the complete frame. More recent research effortstie the classic hybrid coding-structure with motion estimation andcompensation techniques in the wavelet domain, leading to the so calledin-band ME/MC class of wavelet video codecs [12][13][14][16]. This classof codecs presents a conceptually more appealing approach since themultiresolution features of the transform can be used so as to providean inherent spatial and quality scalability similar to wavelet-basedcoding of still images. Hence, if motion compensation is performed in alevel-by-level manner in the wavelet subbands, a decoder can decodewithout drift a video sequence with the horizontal and verticalframe-dimensions having half or quarter-size, since the same informationas the encoder is utilized. In addition, the complexity is reduced,since the inverse wavelet transform is removed from the coding loop.However, a major bottleneck for this approach is that the classicaldyadic wavelet decomposition (named also as the critically-sampledrepresentation) is only periodically shift-invariant [14][16][17], witha period that corresponds to the subsampling factor of the specificdecomposition level. Hence, accurate motion estimation is not feasibleby using only the critically-sampled pyramid. Extensive research effortshave been spent in the recent years to overcome the shift-varianceproblem of the critically sampled wavelet transform. One commonalternative is to use near shift-invariant wavelet transforms, and thereare many solutions in the literature for this type of transforms.However, their main limitation stems from the fact that they all implysome degree of redundancy in comparison to the critically-sampleddecomposition [18][19]. An example of a video-coding scheme thatutilizes such a near shift-invariant transform, namely the complexwavelet transform of Kingsbury [18], is presented in [15]. Theredundancy factor for this transform is four. Although the codingresults obtained with this technique seem promising, the maindisadvantage is that after performing in-band motion estimation/motioncompensation (ME/MC), the error frames contain four times more waveletcoefficients than the input-frame samples. As a consequence, theerror-frame coding tends to be inefficient, thus more complexerror-frame coding algorithms should be envisaged to improve the codingperformance. With this respect, it is important to notice that there isa trade-off between critical sampling implying efficient error-framecoding and redundancy of the transform implying near shift invariance. Acompletely different solution breaking this trade-off, that is,overcoming the shift-variance problem of the DWT while still producingcritically sampled error-frames is the low-band shift method (LBS)introduced theoretically in [16] and used for in-band ME/MC in [14].Firstly, this algorithm reconstructs spatially each reference frame byperforming the inverse DWT. Subsequently, the LBS method is employed toproduce the corresponding overcomplete wavelet representation, which isfurther used to perform in-band ME and MC, since this representation isshift invariant. Basically, the overcomplete wavelet decomposition isproduced for each reference frame by performing the “classical” DWTfollowed by a unit shift of the low-frequency subband of every level andan additional decomposition of the shifted subband. Hence, the LBSmethod effectively retains separately the even and odd polyphasecomponents of the undecimated wavelet decomposition [17]. The“classical” DWT (i.e. the critically-sampled transform) can be seen asonly a subset of this overcomplete pyramid that corresponds to a zeroshift of each produced low-frequency subband, or conversely to theeven-polyphase components of each level's undecimated decomposition. Themotion vectors can be detected by searching directly in the overcompletewavelet representation of the reference frame to find the best match forthe subband information present in the critically-sampled transform ofthe current frame. The motion compensation for the current frame is thenperformed directly in its critically-sampled decomposition. Hence, theproduced error-frames are still critically-sampled. In comparison toimage-domain ME/MC, the in-band ME/MC results of [14] demonstratecompetitive coding performance, especially for high coding-rates.

SUMMARY OF THE INVENTION

The invention provides methods and apparatus for coding and/or encodinga bit stream comprising a representation of a sequence of n-dimensionaldata structures or matrices, in which n is typically 2. A part of thedata of one data structure of the sequence can be mappable withinpredefined similarity criteria to a part of the data of a another datastructure of the sequence. The invention also includes decoders andencoders for coding such a bitstream, e.g., a video signal for use inmotion estimation and/or compensation as well as filter modules fordigital filtering of such bit streams and computer program products forexecuting such filtering as well as subband coding methods. Thesimilarity criteria may include, for instance, a spatial or temporalshift of the data within an image of a video signal such as is used inmotion estimation and/or compensation of moving pictures, e.g., videoimages as well as coders and encoders for coding such a bitstream, e.g.,a video signal for use in motion compensation and/or estimation. Thedata structures are typically video frames and a part of the datastructure may be a block of data within a frame. The shift may be anysuitable shift such as a linear translation at any suitable angle, arotation of the data or change of size such as zooming between a part ofthe data in one data structure and a part of the data in another datastructure of the sequence. The mapping may be to earlier and/or laterdata structures in the sequence. The data structures are typicallysequential frames of a video information stream.

The invention provides a method of digital encoding or decoding adigital bit stream, the bit stream comprising a representation of asequence of n-dimensional data structures, the method being of the typewhich derives at least one further subband of an overcompleterepresentation from a complete subband transform of the data structures,the method comprising:

-   providing a set of one or more critically subsampled subbands    forming a transform of one data structure of the sequence;-   applying at least one digital filter to at least a part of the set    of critically subsampled subbands of the data structure to generate    a further set of one or more subbands of a set of subbands of an    overcomplete representation of the data structure, wherein the    digital filtering step includes calculating at least one further    subband of the overcomplete set of subbands at single rate. In the    method a part of the data of one data structure of the sequence can    be mapped within predefined similarity criteria to a part of the    data of a another data structure of the sequence. The digital filter    may be applied only to members of the set of critically subsampled    subbands of the transform of the data structure. The method may use    a digital filter having at least two non-zero values. The bit stream    may be a video bit stream. The digital subband transform may be a    wavelet. The method may be used in motion compensation and/or motion    estimation of video or other signals which in turn allows    compression of the coded video signals. The motion estimation may be    carried out in the spatial domain or in the subband transform    domain. In the motion estimation a current frame is compared with a    reference frame which may be an earlier or later frame. The result    of the motion estimation is the selection of one or more subbands    from the set of subbands making up the overcomplete representation    which is or are a best approximation to a shifted version of the    reference video frame. This or these selected subbands are then used    for motion compensation. To make the selection all the subbands of    the overcomplete representation may be generated, or alternatively,    if the motion estimation is known, e.g., from the spatial domain,    only the relevant subband or subbands need to be generated.

The invention provides a coder for digital subband coding of a bitstream, the bit comprising a representation of a sequence ofn-dimensional data structures, the coder being of the type which derivesat least one further subband of an overcomplete representation from acomplete subband transform of the data structures, the coder comprising:

-   means for providing a set of one or more critically subsampled    subbands forming a transform of one data structure of the sequence;-   means for applying at least one digital filter to at least a part of    the set of critically subsampled subbands of the data structure to    generate a further set of one or more further subbands of a set of    subbands of an overcomplete representation of the data structure,    wherein the means for applying at least one digital filter includes    means for calculating at least a further subband of the overcomplete    set of subbands at single rate. The coder may be used in motion    compensation and/or motion estimation of video or other signals    which in turn allows compression of the coded video signals. A    motion estimation module may carry out motion estimation in the    spatial domain or in the subband transform domain. In the motion    estimation module means for comparing a current frame with a    reference frame is provided; The reference frame may be an earlier    or later frame. The motion estimation module also comprises means    for selection of one or more subbands from the set of subbands    making up the overcomplete representation which is or are a best    approximation to a shifted version of the reference video frame. A    motion compensation module uses this or these selected subbands for    motion compensation. To make the selection means for generating all    the subbands of the overcomplete representation may be provided, or    alternatively, if the motion estimation is known, e.g., from the    spatial domain, only means for generating the relevant subband or    subbands need to be provided.

In accordance with the invention a decoder may receive data structureswhich are data frames and the set of critically subsampled subbands ofthe transform of the data structure may define a reference frame, andthe decoder further comprises:

means to map a part of the data of one data structure of the sequence toa part of the data of a another data structure of the sequence withinpredefined similarity criteria and to generate a motion vector for thatpart and means to select a further subband of the overcomplete set ofsubbands in accordance with the motion vector. The decoder may furthercomprising means to provide a motion compensated representation of thereference frame using the selected further subband of the overcompleteset of subbands.

The invention also provides a computer program product comprisingexecutable machine readable computer code which executes at least onedigital filter for application to at least a part of a set of criticallysubsampled subbands of a data structure to generate a further set of oneor more further subbands of a set of subbands of an overcompleterepresentation of the data structure, wherein the application of the atleast one digital filter includes calculating at least a further subbandof the overcomplete set of subbands at single rate. The computer programproduct may be stored on a data carrier.

The invention also includes a digital filter module comprising means forapplication of a digital filter to at least a part of a set ofcritically subsampled subbands of a data structure to generate a furtherset of one or more further subbands of a set of subbands of anovercomplete representation of the data structure, wherein theapplication of the at least one digital filter includes calculating atleast a further subband of the overcomplete set of subbands at singlerate.

It is an object of the invention to provide a method and apparatus forperforming a subband transform which is easy to implement than knownmethods and apparatus.

It is also an object of the invention to provide a method and apparatusfor performing a subband transform which requires less calculation stepsthan conventional methods and apparatus.

It is still a further object of the invention to provide a methoddigital filtering and apparatus for digital filtering to generate anovercomplete representation which is easy to implement than knownmethods and apparatus.

It is still a further object of the invention to provide compauterprogram products for carrying out a method for performing a subbandtransform when executed on a computing device.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 a is an encoder and FIG. 1 b is a decoder with which embodimentsof the invention may be used.

FIG. 2 shows a representation of a three-level overcomplete DWTdecomposition.

FIG. 3 shows an example of the derivation of the subbands of theovercomplete wavelet transform of level 3 using the conventional LL-LBSmethod. In a scalable coding framework, only subbands A₀ ³,D₀ ³ areavailable at this level, since subbands D₀ ² and D₀ ¹ cannot be receivedby the decoders of this resolution.

FIG. 4 shows an example of the derivation of the subbands of theOvercomplete Wavelet Transform of level 3 using a prediction-filtermethod in accordance with an embodiment of the invention.

FIG. 5, shown on two pages as FIGS. 5A and 5B, shows an Overcomplete DWTof k+1 levels starting from signal X . A “fictitious” pyramid thatemerges from signal X_(fi) is shown in the circled area.

FIG. 6 shows a memory structure for the concurrent application of theL-tap filters U[n] and W[n] to the input sequence I[n] using a kernelwith L multipliers according to an embodiment of the invention. Thefilters are linked with the relation W(z)=z^(−a)U(z⁻¹).

FIG. 7 shows in the upper half the mirroring at the subband edges andthe initiation and finalization of the inverse transform for level k−1that emerges from level k. The lower half shows the mirroring andinitiation and finalization of the forward transform of level 1 thatemerges from the reconstructed signal X. The figure represents theinverse and forward transforms with the 9/7 filter-pair, whereT_(H)=9,T_(G)=7.

FIG. 8 shows a wavelet video encoder according to an embodiment of theinvention based on ‘bottom-up’ ODWT; motion estimation in spatialdomain.

FIG. 9 shows a wavelet video encoder according to an embodiment of theinvention based on ‘bottom-up’ ODWT; motion estimation in waveletdomain.

FIG. 10 shows a wavelet video decoder in accordance with an embodimentof the invention based on ‘bottom-up’ ODWT.

FIG. 11 shows a wavelet video encoder in accordance with an embodimentof the invention based on ‘bottom-up’ MC; motion estimation in waveletdomain.

FIG. 12 wavelet video decoder in accordance with an embodiment of theinvention based on ‘bottom-up’ MC.

FIG. 13 shows three wavelet decompositions for different translations Tof the input signals X(z).

FIG. 14 shows frame 5 of a football sequenve decomprtessed at 760 kps indifferent resolutions. From top to bottom: original size, halfresolution and quarter resolution.

FIG. 15 shows a schematic representation of a telecommunications systemto which the invention may be applied.

FIG. 16 shows a subband coding circuit in accordance with an embodimentof the invention.

FIG. 17 shows a further subband coding circuit in accordance with anembodiment of the invention.

DEFINITIONS: DWT: Discrete Wavelet Transform SBC: Subband Coder SBD:Subband Decoder EC: Entropy Coder CODWT: Complete-to-Overcomplete DWTME: Motion Estimation MC: Motion Compensation MVs: Motion Vectors MVC:Motion-Vector Coder IDWT: Inverse DWT ED: Entropy Decoder MVD:Motion-Vector Decoder

-   Single rate: a calculating a subband without upsampling or    downsampling.-   Level: refers to a level of a subband pyramid containing the    subbands of the subband transform of a data structure such as an    image-   Level-by-level encoding: in a multiresolutional, multilevel coding    scheme, encoding each level of the subband transformed data    structure to allow transmission of that level (resolution)    independently of other levels (resolutions).-   Level-by-level decoding: in a multiresolutional, multilevel coding    scheme, decoding each level of the received bit stream to allow    display of that level (resolution) independently of other levels    (resolutions).-   Scalability: the ability to decode a coded bitstream to different    resolutions-   Temporal scalability: ability to change the frame rate to number of    frames ratio in a bitstream of framed digital data-   Quality scalability: the ability to change to the quality of a    display-   Overcomplete representation-   Critically-sampled representation: a transform having the same    number of coefficients as the data structure being transformed-   LBS: Low-band shift-   Baseline quality-layer: minimum information to be transmitted to    provide a reconstruction of a data structure in the receiver.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

The invention will be described with reference to certain embodimentsand drawings but is not limited thereto but only by the attached claims.In the following reference is made to subband transforms. Includedwithin these techniques are: wavelet, Discrete Fourier transform, FastFourier transform. In the following the invention will mainly bedescribed with reference to in band motion compensation and estimationof framed data but the invention may find wider application than thisapplication alone. Further, in the following the invention will beillustrated with respect to linear translations between parts of videoframes, but the invention is not limited to linear translation modelsfor motion compensation and estimation.

During encoding of video digital signals an original video imageinformation stream is transformed into another digital representation.This digital representation may be stored on a suitable data carrier ortransmitted to a remote location. The aim of coding is to provide anadvantage, e.g., data compression and/or scalability of the data stream.The aim of decoding such an encoded signal is to reconstruct theoriginal video information as economically and as well as possible,e.g., a lossless reconstruction. The arrays of pixels of such closeframes often contain the same luminance and chrominance informationexcept that the coordinates of pixel positions in the arrays are shiftedor displaced. Shifting in position within the array is a function oftime and defines a motion of these pixels within the arrays. The motionmay be approximated by a motion vector. The encoding is based on thefact that at least parts of temporally close video frames either in theforward or backward direction are often quite similar except for motion.This similarity means that repeat transmission of this moving data ineach frame is not required, it is only necessary to transmit a code of ashift of the data from a previous or subsequent frame, e.g., a motionvector.

Typically, the motion used for motion estimation and compensation is alinear translation, however more complex motions or changes in pixeldistributions can be considered as a basis for motion compensation andestimation. Hence, the invention includes within its scope alternativemotion estimation and compensation models, e.g., linear translations,rotational translation of pixels, zooming of one part of an imagecompared to another. Hence, the model used may be generalized to saythat there is a mapping between a part of one video frame and anothervideo frame, this mapping being determined by predefined similaritycriteria, that is the criteria which are to be used to determine thatthere is similarity between the parts of the two frames. This similaritymay include linear translation, rotational translation, zooming betweenframes.

In one aspect the invention provides an approach for in-band ME/MCwherein prediction-filters are used. The algorithm can achievemathematically the same result as the LBS algorithm of [14]. The LBSalgorithm is suitable for coding and/or encoding a bit stream comprisinga representation of a sequence of n-dimensional data structures, inwhich n is typically 2 or 3, and where a part of the data of one datastructure of the sequence maps within predefined similarity criteria toa part of the data of a another data structure of the sequence and is ofthe type which derives at least one further subband of an overcompleterepresentation from a complete subband transform of the data structures.The use in accordance with embodiments of the invention ofprediction-filters for in-band ME/MC overcomes any shift-varianceproblem of the subband transform such as DWT. Critically samplederror-frames are produced. With this algorithm, the overcomplete waveletrepresentation of each reference frame is produced directly from itscritically sampled wavelet representation by using a set of predictionfilters. In this way, there is no need to reconstruct the spatial domainrepresentation of each reference frame, and to produce from it thecorresponding overcomplete representation, as is the case of the LBSmethod. When used in a level-by-level manner (i.e. ensuring spatialscalability), this is an advantage over the LBS method. Level-by-levelprocessing provides scalability as each level may be transmitted and/orreceived and processed independently. Fast level-by-level calculation ofthe overcomplete discrete wavelet transform starting from the criticallysampled pyramid is disclosed including a generalized form for theprediction filters for an arbitrary number of decomposition levels. Somesymmetry properties between the prediction filters are disclosed,specifically, for biorthogonal point-symmetric filter-pairs that are themost-performing types of wavelets in still-image coding, leading toefficient implementation. A comparison from the computational point ofview of the invention with the LBS method is given.

An example of a communication system 210 which can be used with theinvention is shown in FIG. 15. It comprises a source 200 of information,e.g., a source of video signals such as a video camera or retrieval froma memory. The signals are encoded in an encoder 202 resulting in a bitstream, e.g., a serial bit stream which is transmitted through a channel204, e.g., a cable network, a wireless network, an air interface, apublic telephone network, a microwave link, a satellite link. Theencoder 202 forms part of a transmitter or transceiver if both transmitand receive functions are provided. The received bit stream is thendecoded in a decoder 206 which is part of a receiver or transceiver. Thedecoding of the signal may provide at least one of spatial scalablity,e.g., different resolutions of a video image are supplied to differentend user equipments 207-209 such as video displays; temporalscalability, e.g., decoded signals with different frame rate/framenumber ratios are supplied to different user equipments; and qualityscalability, e.g., decoded signals with different signal to noise ratiosare supplied to different user equipments.

A potential architecture of a wavelet-based video codec performingin-band motion estimation/motion compensation (ME/MC) with which a firstembodiment of the invention may be used is given in FIG. 1 whereby FIG.1 a shows the encoder and FIG. 1 b the decoder. This architecture can beseen as an extension of the “classical” architecture of transform-basedvideo coders/decoders. In this scheme there are two modes of operationas determined by the setting of the switches 16, 18, as explained in thefollowing. In the intra-frame coding mode, the current input frame 4 isfirst of all wavelet transformed using a suitable compression techniquesuch as the DWT module 2 of FIG. 1, the resulting wavelet subbands arequantized and encoded using an embedded intra-band coding technique,(SBC module 8) and the result is entropy-coded using the EC module 10.The embedded intra-band coding and the entropy coding are performed in alevel-by-level fashion to guarantee resolution scalability. Furthermore,this process is performed until the target bit-rate for the currentlycoded frame is met. The second coding mode is the inter-frame codingmode, where each reference frame is first of all wavelet transformedusing the DWT module 2 of FIG. 1. As in the previous mode, the subbandsof every decomposition level are encoded using the intra-band embeddedcoding technique in SBC module 8 and entropy-coded in EC module 10. TheSBD module 12 performs the decoding operation and produces the basequality layer of the reference frame, which is stored in a memory buffer20. The complete-to-overcomplete DWT module 22 (CODWT) produces, foreach level, the set of one or more subbands of the overcompleterepresentation of the reference frame starting from the subbands of thecritically sampled pyramid. These are stored in a buffer memory 24. Theresulting n subbands per level are then used during the motionestimation which is performed by the block-based ME module 30 to findthe best match between the subband-information present in the currentframe, received at module 30 from the output of DWT module 2, and theovercomplete representation of the reference frame received from thebuffer 24. A block 26 is selected using selecting means in the motioncompensation module 28 or the motion estimation module 30 from thebuffered subbands of the reference frame that represents the best matchis used for the motion-compensation process which is performed by the MCmodule 28. The resulting motion-vectors of every decomposition level aresent to the motion-vector coder (MVC module 32). The output of themotion compensation is an error frame in the wavelet domain which isgenerated by subtracting the output of MC module 28 from the subbandtransformed current frame in subtractor 6. This error frame isintra-band embedded coded in SBC module 8 and entropy-coded in EC module10 in a level-by-level fashion. To generate the reference frame 20 fromthe subband decoded intra-band coded signal the output of SBD 12 isadded to the output from MC 28 in adder 14. This reference frame is usedin the next iteration. The dashed line shows a less preferred embodimentin which the error frame is supplied directly to the adder 14.

As shown in FIG. 1 b, the decoder operates exactly in the mirrorfashion, the intra-frame or inter-frame decoding being determined by theswitch 52. Hence, in the intra-frame mode, the resolution-scalable codedframe is received and the bit-stream parsing can cease at any resolutionor quality level (provided that the minimal base-quality layer isreceived). The next operations are entropy decoding in ED module 32followed by subband decoding in SBD module 34 and inverse DWT in IDWTmodule 40. At this stage, the frame corresponding to the specificoperational settings e.g., quality and resolution, is reconstructed.

In the inter-frame mode, the error-frame is received at the desiredquality level at the ED module 32 and the corresponding motion vectorsof each decomposition level are received and decoded as well by a MVDmodule 50. Similar to the encoder, the complete-to-overcomplete DWT ofthe reference frame 46 is constructed in a level-by-level manner inCODWT module 48. The key difference is that since the motion vectors areknown, only individual blocks 44 needed for the motion compensationwhich is to be performed in module 42, are calculated by the CODWTmodule 48. In the motion-compensation phase carried out in module 42,these blocks 44 are used to reconstruct the predicted frame. Thepredicted frame is added to the error-frame 36 of the currentdecomposition level in adder 38. To ensure drift-free qualityscalability, the CODWT unit 48 operates on the base-quality layer forevery decomposition level. Hence the decoder reconstructs the identicalovercomplete representation per level as the encoder. In this manner,the decoder is free to stop decoding at any resolution or quality level.

From the description of the previous subsection, it can be seen that theCODWT module 22, 48 is a key part of the ME/MC subsystem of the scalablevideo-codec.

FIG. 2 shows an example of the 1-D overcomplete pyramid for threedecomposition levels as constructed by the LBS method. Initially, theinput signal X is decomposed into two sets by retaining separately theeven or odd polyphase components of the undecimated decomposition, orequivalently performing two “classical” wavelet decompositions; one inthe zero-shifted input signal and one in the unit-shifted input. Each ofthe low-frequency subbands of the two decompositions is further analyzedinto two sets by performing again an undecimated wavelet decompositionand retaining separately the even and odd polyphase components and soon. The 2-D overcomplete pyramid is constructed in the identical manner,with the application of the LBS method in the input-subband rows and thecolumns of the results. Hence, to facilitate the explanation, thedescription in the following is restricted to the one-dimensional casefor clarity's sake, with the extension in two dimensions following therow-column approach. By definition S₀, S₁ denote the even and oddpolyphase components of signal S respectively (named also type 0 andtype 1). Each low-frequency (average) subband of decomposition level iis denoted as A_(x) ^(i), where x is a binary representation that showsthe subband type, with the Most-Significant Bit (MSB) denoting the“Mother”—subband type and the Least-Significant Bit (LSB) denoting thecurrent-subband type. For example, subband A_(&011) ³ denotes thelow-frequency subband of level 3 that has been produced by retaining theeven polyphase components of decomposition level 1 and the odd polyphasecomponents of levels 2 and 3. Similar notations apply for thehigh-frequency (detail) subbands D_(x) ¹. Note how the LSB of the binaryrepresentation denotes a “turn” in the pyramid (0 for “left”, 1 for“right”) in comparison to the previous-level subband (“parent”). Thus,for every decomposition level, the subscript bits are shifted to theleft and a new LSB is entered, depending on the polyphase grid that isretained from the decomposition (even, odd). In a scalable-codingframework, the key-difference is that the subband-transmission anddecoding occur in a bottom-up manner, every decoder receives thecoarsest-resolution subbands first (i.e. subbands A₀ ³,D₀ ³) and is freeto stop the bitstream-parsing at any time after the baselinequality-layer has been received for each level. In this way, the qualityand resolution of the decoded video can vary accordingly. Under such aprogressive decoding environment, the LBS method is modified to performa level-by-level construction of the overcomplete representation(denoted by LL-LBS), starting from the subbands of thecritically-sampled transform of each coded level. Such a situation isillustrated in FIG. 3 for the LL-LBS method applied on a 3-leveldecomposition; firstly, three inverse wavelet transforms are performedstarting from the subbands A₀ ³,D₀ ³, as shown in the left part of FIG.3. Notice that in this situation, the subbands D₀ ² and D₀ ¹ are notpresent (i.e. are not received at the decoder site), and as aconsequence, the subbands shown in FIG. 3 are not identical with thecorresponding ones of FIG. 2. Subsequently, from the highest-resolutionsignal X, all the subbands A_(i) ³,D_(i) ³, iε[1,7],iεZ are constructedby performing the forward transforms shown in the rest of FIG. 3. Thelow-frequency subbands A_(i) ^(k) are constructed only if k is thecoarsest-resolution level of the decomposition. In all the other cases,only the high-frequency subbands D_(j) ^(k), with jε[1,2^(k)−1], jεZ areneeded. This is imposed by the fact that no motion estimation orcompensation is performed for the subband A₀ ^(k) if k is not thecoarsest decomposition level. Hence, in the example if after theconstruction of the overcomplete transform of level 3, one moreresolution level is received (i.e. subband D₀ ²), then the LL-LBSoperates again in the same fashion, but constructs only the subbandsD_(j) ², with jε[1, 3], jεZ. Although approximations of the subbandsD_(j) ² can already be calculated during the calculations of level 3, atthis time the subband D₀ ² is not available; as a consequence, theseapproximations are obtained based only on the subband A₀ ² and hencethey do not have the best accuracy possible. This observation indicatesa significant difference between the LL-LBS and the LBS methods: becauseof the bottom-up level-by-level construction, the high-frequencysubbands of the higher-resolution levels (levels 2 and 1 in the example)are not available when the current level is processed (level 3). Hence,the resulting overcomplete representation of each level obtained withthe LL-LBS method is not identical to the one constructed with the LBSalgorithm. Actually, in its original form, the LBS algorithm creates theovercomplete representation under the assumption that all the subbandsof the critically sampled pyramid are available, and this is not alwaysthe case in a resolution-scalable framework. This means that the LBSmethod requires the receipt of all resolutions in order to create theovercomplete representation. This places a limitation on the timerequired to decode an image—all resolutions should be available.However, once all the information has been made available at thedecoding time, the level-by-level LBS produces the subbands of theovercomplete representation with the best accuracy possible whilesimultaneously ensuring drift-free, full resolution-progressive decoding(i.e. spatial scalability).

In accordance with one embodiment of the invention, it is not necessaryto receive the highest resolution information in order to generate auseful overcomplete representation. In accordance with an aspect of theinvention, the overcomplete representation can be obtained by theapplication of prediction filters. An example of the derivation of thesubbands of level 3 with the prediction-filters method is given in FIG.4. It can be noticed that the overcomplete representation is “predicted”in a level-by-level manner using the sets of filters F_(R) ^(Q), withQε[1,3],QεZ and Rε[0,15],RεZ indicated under the subband-pairs of FIG.4. However, no upsampling or downsampling is performed with thisalgorithm and this leads to substantial complexity reductions. Where noupsampling or downsampling is used, the calculation is said to be atsingle rate. The form and utilization of filters F_(R) ^(Q) is explainedin the following, where, in accordance with an embodiment of theinvention, the general proof for an arbitrary-levelcomplete-to-overcomplete transform is presented. In the same manner asin the LL-LBS method, for the higher-resolutions (decomposition levels 2and 1 in the example of FIG. 3), only the high-frequency subbands areconstructed. Also, if applied in a level-by-level manner, both methodsproduce identical results. In the general case of an overcompleterepresentation with k+1 levels shown in FIG. 5, the binaryrepresentation of the subband indices becomes impractical, and thus theyare simply denoted with their decimal equivalent, in the form

${\sum_{i = 0}^{k}\;{b_{i}2^{i}}},$with b_(i)={0,1}. In addition, as shown in FIG. 2, the overcompletepyramid is separated into the “left-half” and “right-half” pyramidsrespectively. These parts correspond to the two “Mother” subbandscontaining the even and odd-polyphase components respectively of theundecimated decomposition of the original signal. To conclude thenotations, in the Z-plane expressions, frequently H(z), A₄ ³(z), F₂ ¹(z)are simply denoted as H ,A₄ ³,F₂ ¹ respectively, to reduce theexpressions' length, while (H)₀,(H)₁ denote the even or odd polyphasecomponents (similar applies for H_(i,0),H_(i,1) for filter H_(i)).

In this section a generic framework is presented for an embodiment of aprediction filter. This framework allows the complete to overcompletetransform derivation for an arbitrary decomposition level. Themathematical derivations are performed for the 1-D overcompleterepresentation; nevertheless, as it will be shown later, the extensionto 2-D is straightforward following the row-column approach orequivalent thereto. Firstly, the subbands of decomposition level k ofthe overcomplete representation are expressed as a function of thecritically-sampled wavelet decomposition (i.e. the subbands A₀ ^(k),D₀^(l), with lε[1,k],lεZ). From this generic mathematical formulation, thelevel-by-level overcomplete transform of level k is readily extracted asa special case, in which the overcomplete subbands are calculated usingonly the critically-sampled representation of the same level. Finally,the symmetry properties of the prediction filters for everydecomposition level are given, which allow their efficientimplementation.

Derivation of Subbands of Decomposition Level k from theCritically-Sampled Pyramid of Level k—the Prediction Filters inAccordance with an Embodiment of the Invention

The proof of the general form of the invented prediction filters is asfollows in which the prediction filters for the decomposition levelsE=1,2,3 are derived. The proposition P(1) corresponding to E=1 is:

$\begin{matrix}{{P(1)}:\left\{ {\begin{matrix}{A_{1}^{1} = {{F_{0}^{1}A_{0}^{1}} + {F_{1}^{1}D_{0}^{1}}}} \\{D_{1}^{1} = {{F_{2}^{1}A_{0}^{1}} + {F_{3}^{1}D_{0}^{1}}}}\end{matrix}.} \right.} & (1)\end{matrix}$For level E=2, the set of prediction propositions P(2) is:

$\begin{matrix}{{P(2)}:{\begin{matrix}\left\{ \begin{matrix}{A_{1}^{2} = {{F_{0}^{1}A_{0}^{2}} + {F_{1}^{1}D_{0}^{2}}}} \\{D_{1}^{2} = {{F_{2}^{1}A_{0}^{2}} + {F_{3}^{1}D_{0}^{2}}}}\end{matrix} \right. \\\left\{ \begin{matrix}{A_{2}^{2} = {{F_{0}^{2}A_{0}^{2}} + {F_{1}^{2}D_{0}^{2}} + \left( {{H \cdot F_{1}^{1}}D_{0}^{1}} \right)_{0}}} \\{D_{2}^{2} = {{F_{2}^{2}A_{0}^{2}} + {F_{3}^{2}D_{0}^{2}} + \left( {{G \cdot F_{1}^{1}}D_{0}^{1}} \right)_{0}}}\end{matrix} \right. \\\left\{ \begin{matrix}{A_{3}^{2} = {{F_{4}^{2}A_{0}^{2}} + {F_{5}^{2}D_{0}^{2}} + \left( {{H \cdot F_{1}^{1}}D_{0}^{1}} \right)_{1}}} \\{D_{3}^{2} = {{F_{6}^{2}A_{0}^{2}} + {F_{7}^{2}D_{0}^{2}} + \left( {{G \cdot F_{1}^{1}}D_{0}^{1}} \right)_{1}}}\end{matrix} \right.\end{matrix}.}} & (2)\end{matrix}$The proofs for E=1 and E=2 are given in Appendix I. The generalizationand the proof for an arbitrary level E=k will be attempted withmathematical induction. Thus we assume the set of predictionpropositions P(k) for an arbitrary level k to be true and based on themwe derive the propositions P(k+1) and the filters for level k+1. Theformulas derived for level k+1 are true if, and only if, they can bederived from level k by replacing k with k+1. In addition, if they aretrue for level k+1, then according to the induction principle, they aretrue for any level E.

Therefore, let us assume that the propositions for the levels E=1,2 , .. . k−1, k with k≧2 are all true. The proposed set of propositions P(k)corresponding to the level E=k of the overcomplete decomposition pyramidis:

$\begin{matrix}{{P(k)}:\left\{ \begin{matrix}{A_{x}^{k} = {{F_{4p}^{l + 1}A_{0}^{k}} + {F_{{4p} + 1}^{l + 1}D_{0}^{k}} + \left( {{H \cdot F_{i{(1)}}^{l}}D_{0}^{k - 1}} \right)_{b_{0}} + \left\lbrack {H\left( {{H \cdot F_{i{(2)}}^{l - 1}}D_{0}^{k - 2}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + \left\lbrack {H\left( {{H\left( {\ldots\left( {{H \cdot F_{i{(l)}}^{1}}D_{0}^{k - 1}} \right)}_{b_{l - 1}} \right)}_{b_{l - 2}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}} \\{D_{x}^{k} = {{F_{{4p} + 2}^{l + 1}A_{0}^{k}} + {F_{{4p} + 3}^{l + 1}D_{0}^{k}} + \left( {{G \cdot F_{i{(1)}}^{l}}D_{0}^{k - 1}} \right)_{b_{0}} + \left\lbrack {G\left( {{H \cdot F_{i{(2)}}^{l - 1}}D_{0}^{k - 2}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + \left\lbrack {G\left( {{H\left( {\ldots\left( {{H \cdot F_{i{(l)}}^{1}}D_{0}^{k - 1}} \right)}_{b_{l - 1}} \right)}_{b_{l - 2}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}}\end{matrix} \right.} & (3)\end{matrix}$A pictorial representation of the subbands of level k and thecorresponding prediction filters is given in FIG. 5. In these equations,xε[1,2^(k)−1], xεZ denotes the subband index at level k, and it iswritten as:x=2^(l) +p,  (4)where l is given by:l=└log ₂ x┘,  (5)and └a┘ denotes the integer part of a. From (4) and (5) if follows thatlε[0,k−1],lεZ and pε[0,2^(l)−1], pεZ for any subband index x taken inthe interval [1, 2^(k)−1]. Considering l≧1 in the set of equations (3),b_(i), 0≦i≦l−1 are defined as the binary values from the binaryrepresentation of p, given by

${p = {\sum_{i = 0}^{l - 1}\;{b_{i} \cdot 2^{i}}}},$with b_(i)ε{0,1}, while i(m) are filter indices defined as i(m)

${{i(m)} = {{4\left\lfloor \frac{p}{2^{m}} \right\rfloor} + 1}},$for mε[1,l], mεZ. In the particular case of l=0 corresponding to k=1 andx=1 we set b_(i)=0, ∀iεZ and F_(A) ^(B)=0 for any indices A<0 or B≦0, toensure that the set of equations (3) is identical with P(1) given in(1).

The proposed set of propositions P(k) can be divided into two sets ofequations P_(L)(k) and P_(R)(k) with P(k)=P_(L)(k)∪P_(R)(k), each setcorresponding to the subbands of the “left-half” and “right-half” partsrespectively of the pyramid of level k (see FIG. 3). Proving byinduction the set of propositions P(k) is equivalent to proving byinduction each set of propositions P_(L)(k) and P_(R)(k).

The set of propositions P_(L)(k) corresponding to the left side of thepyramid of level k, with k≧2 is obtained from (3) for xε[1,2^(k−1)−1]:

$\begin{matrix}{{P_{L}(k)}:\left\{ \begin{matrix}{A_{x}^{k} = {{F_{4p}^{l + 1}A_{0}^{k}} + {F_{{4p} + 1}^{l + 1}D_{0}^{k}} + \left( {{H \cdot F_{i{(1)}}^{l}}D_{0}^{k - 1}} \right)_{b_{0}} + \left\lbrack {H\left( {{H \cdot F_{i{(2)}}^{l - 1}}D_{0}^{k - 2}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + \left\lbrack {H\left( {{H\left( {\ldots\left( {{H \cdot F_{i{(l)}}^{1}}D_{0}^{k - 1}} \right)}_{b_{l - 1}} \right)}_{b_{l - 2}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}} \\{D_{x}^{k} = {{F_{{4p} + 2}^{l + 1}A_{0}^{k}} + {F_{{4p} + 3}^{l + 1}D_{0}^{k}} + \left( {{G \cdot F_{i{(1)}}^{l}}D_{0}^{k - 1}} \right)_{b_{0}} + \left\lbrack {G\left( {{H \cdot F_{i{(2)}}^{l - 1}}D_{0}^{k - 2}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + \left\lbrack {G\left( {{H\left( {\ldots\left( {{H \cdot F_{i{(l)}}^{1}}D_{0}^{k - 1}} \right)}_{b_{l - 1}} \right)}_{b_{l - 2}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}}\end{matrix} \right.} & (6)\end{matrix}$Note that for k=1 there are no predictions to be made for the left sideof the pyramid, that is P_(L)(1)=Ø. The remaining set of propositionsP_(R)(k) corresponding to the right side of the pyramid of level k isobtained for xε[2^(k−1), 2^(k)−1]. Notice that from (5) it results thatl=k−1 therefore equation (4) is equivalent to:x=2^(k−1) +p.  (7)Since xε[2^(k−1), 2^(k)−1], it results from (7) that pε[0, 2^(k−1)−1],pεZ. Replacing l=k−1 in (3) yields:

$\begin{matrix}{{P_{R}(k)}:\left\{ \begin{matrix}{A_{x}^{k} = {{F_{4p}^{k}A_{0}^{k}} + {F_{{4p} + 1}^{k}D_{0}^{k}} + \left( {{H \cdot F_{i{(1)}}^{k - 1}}D_{0}^{k - 1}} \right)_{b_{0}} + \left\lbrack {H\left( {{H \cdot F_{i{(2)}}^{k - 2}}D_{0}^{k - 2}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + \left\lbrack {H\left( {{H\left( {\ldots\left( {{H \cdot F_{i{({k - 1})}}^{1}}D_{0}^{1}} \right)}_{b_{k - 2}} \right)}_{b_{k - 3}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}} \\{D_{x}^{k} = {{F_{{4p} + 2}^{k}A_{0}^{k}} + {F_{{4p} + 3}^{k}D_{0}^{k}} + \left( {{G \cdot F_{i{(1)}}^{k - 1}}D_{0}^{k - 1}} \right)_{b_{0}} + \left\lbrack {G\left( {{H \cdot F_{i{(2)}}^{k - 2}}D_{0}^{k - 2}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + \left\lbrack {G\left( {{H\left( {\ldots\left( {{H \cdot F_{i{({k - 1})}}^{1}}D_{0}^{1}} \right)}_{b_{k - 2}} \right)}_{b_{k - 3}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}}\end{matrix} \right.} & (8)\end{matrix}$We follow in these equations the same definitions as given in equation(3), that is, b_(i), 0≦i≦k−2 with k≧2 are the binary values(b_(i)ε{0,1}) from the binary representation of p given by p=

${p = {\sum_{i = 0}^{l - 1}\;{b_{i} \cdot 2^{i}}}},$and i(m) are filter indices given by i(m)

${{i(m)} = {{4\left\lfloor \frac{p}{2^{m}} \right\rfloor} + 1}},$mε[1,k−1]. In the particular case of k=1, we set b_(i)=0,∀iεZ and F_(A)^(B)=0 for any indices A<0 or B≦0, to ensure that P_(R)(1) given in (8)is identical with P(1) given in (1).

For the particular case of biorthogonal point-symmetric filter-pairs,the filters F_(i) ^(k) of decomposition level k needed to calculate thesubbands of the “right” side of the pyramid of level k , with i=0, 1, .. . , 2^(k−2)−1, are:

$\begin{matrix}{{F_{8i}^{k} = {F_{{4i},0}^{k - 1} - {z^{- 1}F_{3}^{1}F_{{4i},1}^{k - 1}}}},} & (9) \\{{F_{{8i} + 1}^{k} = {z^{- 1}F_{1}^{1}F_{{4i},1}^{k - 1}}},} & (10) \\{{F_{{8i} + 2}^{k} = {z^{- 1}F_{2}^{1}F_{{4i},1}^{k - 1}}},} & (11) \\{{F_{{8i} + 3}^{k} = {F_{{4i},0}^{k - 1} + {z^{- 1}F_{3}^{1}F_{{4i},1}^{k - 1}}}},} & (12) \\{{F_{{8i} + 4}^{k} = {F_{{4i},1}^{k - 1} + {F_{0}^{1}F_{{4i},0}^{k - 1}}}},} & (13) \\{{F_{{8i} + 5}^{k} = {F_{1}^{1}F_{{4i},0}^{k - 1}}},} & (14) \\{{F_{{8i} + 6}^{k} = {F_{2}^{1}F_{{4i},0}^{k - 1}}},} & (15) \\{{F_{{8i} + 7}^{k} = {F_{{4i},1}^{k - 1} + {F_{3}^{1}F_{{4i},0}^{k - 1}}}},} & (16)\end{matrix}$For completeness, we give the filters of the proposition P(1). They havethe form:F ₀ ¹ =Det ⁻¹(H ₁ G ₁ −zH ₀ G ₀), F ₁ ¹ =Det ⁻¹(zH ₀ H ₀ −H ₁ H₁),  (17)F ₂ ¹ =Det ⁻¹(G ₁ G ₁ −zG ₀ G ₀), F ₃ ¹ =Det ⁻¹(zH ₀ G ₀ −H ₁ G₁),  (18)where Det is the determinant of the analysis polyphase matrix H _(p)(z),given by:

$\begin{matrix}{{{{\underset{\_}{H}}_{p}(z)} = \begin{pmatrix}{H_{0}(z)} & {H_{1}(z)} \\{G_{0}(z)} & {G_{1}(z)}\end{pmatrix}},} & (19)\end{matrix}$Let us prove now that the propositions P_(L)(k+1) formulated for levelE=k+1 are true. Notice from FIG. 5 that the “left” side of the pyramidwith k+1 decomposition levels (original pyramid) emerges only from thesubband A₀ ^(i). Hence, one can define a new, “fictitious” pyramid withsubbands A_(fi,j) _(z) ^(i) ^(z) , D_(fi,j) _(z) ^(i) ^(z) ,i_(s)ε[1,k], j_(s)ε[1,2^(i,)−1] that emerges from an input signal X_(fi)with X_(fi)=A₀ ^(i). This pyramid is shown in FIG. 5 in the dashedline-enclosed area. One can notice that the entire decomposition leveli_(s) of this “fictitious” pyramid is equivalent to the “left” side oflevel i_(s+)1 of the original pyramid emerging from X, with therelationship:

$\begin{matrix}{{A_{j_{s}}^{i_{s} + 1} = A_{{fi},j_{s}}^{i_{s}}},{D_{j_{s}}^{i_{s} + 1} = D_{{fi},j_{s}}^{i_{s}}},{i_{s} \in \left\lbrack {1,k} \right\rbrack},{j_{s} \in {\left\lbrack {1,{2^{i_{s}} - 1}} \right\rbrack.}}} & (20)\end{matrix}$Notice that i_(s)ε[1,k], therefore the set of propositions P(k) can beapplied for the level k of the “fictitious” pyramid, since they are trueby assumption. Hence, by using equation (3) one can write:

$\begin{matrix}{\begin{matrix}{A_{{fi},x}^{k} = {{F_{4p}^{l + 1}A_{{fi},0}^{k}} + {F_{{4p} + 1}^{l + 1}D_{{fi},0}^{k}} + \left( {{H \cdot F_{i{(1)}}^{l}}D_{{fi},0}^{k - 1}} \right)_{b_{0}} + \left\lbrack {H\left( {{H \cdot F_{i{(2)}}^{l - 1}}D_{{fi},0}^{k - 2}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots +}} \\{+ \left\lbrack {H\left( {{H\left( {\ldots\left( {{H \cdot F_{i{(l)}}^{1}}D_{{fi},0}^{k - l}} \right)}_{b_{l - 1}} \right)}_{b_{l - 2}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}\end{matrix},} & (21) \\{\begin{matrix}{D_{{fi},x}^{k} = {{F_{{4p} + 2}^{l + 1}A_{{fi},0}^{k}} + {F_{{4p} + 3}^{l + 1}D_{{fi},0}^{k}} + \left( {{G \cdot F_{i{(1)}}^{l}}D_{{fi},0}^{k - 1}} \right)_{b_{0}} + \left\lbrack {G\left( {{H \cdot F_{i{(2)}}^{l - 1}}D_{{fi},0}^{k - 2}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots +}} \\{+ \left\lbrack {G\left( {{H\left( {\ldots\left( {{H \cdot F_{i{(l)}}^{1}}D_{{fi},0}^{k - l}} \right)}_{b_{l - 1}} \right)}_{b_{l - 2}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}\end{matrix},} & (22)\end{matrix}$in which all indices have the same definitions as in equation (3).

By performing the transformations of variables specified by equation(20), equations (21)-(22) become:

$\begin{matrix}{A_{x}^{k + 1} = {{F_{4p}^{l + 1}A_{0}^{k + 1}} + {F_{{4p} + 1}^{l + 1}D_{0}^{k + 1}} + \left( {{H \cdot F_{i{(1)}}^{l}}D_{0}^{k}} \right)_{b_{0}} + \left\lbrack {H\left( {{H \cdot F_{i{(2)}}^{l - 1}}D_{0}^{k - 1}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + \left\lbrack {H\left( {{H\left( {\ldots\left( {{H \cdot F_{i{(l)}}^{1}}D_{0}^{k + 1 - l}} \right)}_{b_{l - 1}} \right)}_{b_{l - 2}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}} & (23) \\{D_{x}^{k + 1} = {{F_{{4p} + 2}^{l + 1}A_{0}^{k + 1}} + {F_{{4p} + 3}^{l + 1}D_{0}^{k + 1}} + \left( {{G \cdot F_{i{(1)}}^{l}}D_{0}^{k}} \right)_{b_{0}} + \left\lbrack {G\left( {{H \cdot F_{i{(2)}}^{l - 1}}D_{0}^{k - 1}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + \left\lbrack {G\left( {{H\left( {\ldots\left( {{H \cdot F_{i{(l)}}^{1}}D_{0}^{k + 1 - l}} \right)}_{b_{l - 1}} \right)}_{b_{l - 2}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}} & (24)\end{matrix}$Notice that these equations are identical to the set of equations (6) inwhich k is replaced with k+1. This means that the propositionsP_(L)(k+1) are true. Additionally, only filters of levels 1,2, . . . , kare used, which are true by the assumption of level k.The propositions P_(R)(k+1) for the “right” side of the pyramid remainto be proven. In addition the form of the filters

F_(i)^(k + 1)has to be proven as well.

The set of propositions P_(R)(k+1) are formulated by replacing k withk+1 in equation (8):

$\begin{matrix}{{P_{R}\left( {k + 1} \right)}:\left\{ \begin{matrix}{A_{x}^{k + 1} = {{F_{4p}^{k + 1}A_{0}^{k + 1}} + {F_{{4p} + 1}^{k + 1}D_{0}^{k + 1}} + \left( {{H \cdot F_{i{(1)}}^{k}}D_{0}^{k}} \right)_{b_{0}} + \left\lbrack {H\left( {{H \cdot F_{i{(2)}}^{k - 1}}D_{0}^{k - 1}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + \left\lbrack {H\left( {{H\left( {\ldots\left( {{H \cdot F_{i{(k)}}^{1}}D_{0}^{1}} \right)}_{b_{k - 1}} \right)}_{b_{k - 2}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}} \\{D_{x}^{k + 1} = {{F_{{4p} + 2}^{k + 1}A_{0}^{k + 1}} + {F_{{4p} + 3}^{k + 1}D_{0}^{k + 1}} + \left( {{G \cdot F_{i{(1)}}^{k}}D_{0}^{k}} \right)_{b_{0}} + \left\lbrack {G\left( {{H \cdot F_{i{(2)}}^{k - 1}}D_{0}^{k - 1}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + \left\lbrack {G\left( {{H\left( {\ldots\left( {{H \cdot F_{i{(k)}}^{1}}D_{0}^{1}} \right)}_{b_{k - 1}} \right)}_{b_{k - 2}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}}}\end{matrix} \right.} & (25)\end{matrix}$To prove these propositions, let us start by performing an inversewavelet transform in order to calculate the A₀ ^(k) subband in functionof the

A₀^(k + 1)  and  D₀^(k + 1)subbands:

$\begin{matrix}{A_{0}^{k} = {{z\left\lbrack {{{\overset{\sim}{H}(z)}{A_{0}^{k + 1}\left( z^{2} \right)}} + {{\overset{\sim}{G}(z)}{D_{0}^{k + 1}\left( z^{2} \right)}}} \right\rbrack}.}} & (26)\end{matrix}$Since both A₀ ^(k) and D₀ ^(k) subbands are known now, we can apply anyof the propositions of level k (involving the F_(i) ^(k) filters), sincethey are true by assumption. Hence, we can calculate any subband A_(x)^(k), with xε[2^(k−1),2^(k)−1] by using the set of propositions P_(R)(k)given in equation (8). By replacing (26) in (8) we obtain:

$\begin{matrix}{{{{A_{x}^{k}(z)} = {{{{zF}_{4p}^{k}(z)}{\overset{\sim}{H}(z)}{A_{0}^{k + 1}\left( z^{2} \right)}} + \;{{{zF}_{4p}^{k}(z)}{\overset{\sim}{G}(z)}{D_{0}^{k + 1}\left( z^{2} \right)}} + {{F_{{4p} + 1}^{k}(z)}{D_{0}^{k}(z)}} + {T\left( {{k - 1},z} \right)}}},}\;} & (27)\end{matrix}$with:

$\begin{matrix}{{{T\left( {{k - 1},z} \right)} = {\left( {{H \cdot F_{i{(1)}}^{k - 1}}D_{0}^{k - 1}} \right)_{b_{0}} + \left\lbrack {H\left( {{H \cdot F_{i{(2)}}^{k - 2}}D_{0}^{k - 2}} \right)}_{b_{i}} \right\rbrack_{b_{0}} + \ldots + \left\lbrack {H\left( {{H\left( {\ldots\left( {{H \cdot F_{i{({k - 1})}}^{1}}D_{0}^{1}} \right)}_{b_{k - 2}} \right)}_{b_{k - 3}}\ldots} \right)}_{b_{i}} \right\rbrack_{b_{0}}}},} & (28)\end{matrix}$and b_(i), i(m) defined as for equation (8). The “tail” T(k−1,z) denotesthe contributions of levels 1,2, . . . , k−1.

In order to calculate the

A_(2x)^(k + 1), D_(2x)^(k + 1)subbands (even-numbered subbands of level k+1), we need to perform asingle-level forward transform, retaining the even samples (“classic”transform):

$\begin{matrix}{\begin{bmatrix}A_{2x}^{k + 1} \\D_{2x}^{k + 1}\end{bmatrix} = {{{\frac{1}{2}\begin{bmatrix}{{H_{0}(z)} + {z^{- \frac{1}{2}}{H_{1}(z)}}} & {{H_{0}(z)} - {z^{- \frac{1}{2}}{H_{1}(z)}}} \\{{G_{0}(z)} + {z^{- \frac{1}{2}}{G_{1}(z)}}} & {{G_{0}(z)} - {z^{- \frac{1}{2}}{G_{1}(z)}}}\end{bmatrix}}\begin{bmatrix}{A_{x}^{k}\left( z^{\frac{1}{2}} \right)} \\{A_{x}^{k}\left( {- z^{\frac{1}{2}}} \right)}\end{bmatrix}}.}} & (29)\end{matrix}$To derive the terms in (29) we need to evaluate A_(x) ^(k)(z^(1/2)) andA_(x) ^(k)(−z^(1/2)). By replacing z with z^(1/2) in (27) we get:

$\begin{matrix}{{A_{x}^{k}\left( z^{\frac{1}{2}} \right)} = {{z^{\frac{1}{2}}{F_{4p}^{k}\left( z^{\frac{1}{2}} \right)}{\overset{\sim}{H}\left( z^{\frac{1}{2}} \right)}{A_{0}^{k + 1}(z)}} + {z^{\frac{1}{2}}{F_{4p}^{k}\left( z^{\frac{1}{2}} \right)}{\overset{\sim}{G}\left( z^{\frac{1}{2}} \right)}{D_{0}^{k + 1}(z)}} + {{F_{{4p} + 1}^{k}\left( z^{\frac{1}{2}} \right)}{D_{0}^{k}\left( z^{\frac{1}{2}} \right)}} + {T\left( {{k - 1},z^{\frac{1}{2}}} \right)}}} & (30)\end{matrix}$The filter

$F_{4p}^{k}\left( z^{\frac{1}{2}} \right)$can be written using the Type I polyphase representation as:

$\begin{matrix}{{{F_{4p}^{k}\left( z^{\frac{1}{2}} \right)} = {{F_{{4p},0}^{k}(z)} + {z^{- \frac{1}{2}}{F_{{4p},1}^{k}(z)}}}},} & (31)\end{matrix}$while {tilde over (H)}(z^(1/2)) and {tilde over (G)}(z^(1/2)) can besubstituted by the Type II polyphase representation, as follows:{tilde over (H)}(z ^(1/2))={tilde over (H)} ₁(z)+z ^(−1/2) {tilde over(H)} ₀(z)  (32){tilde over (G)}(z ^(1/2))={tilde over (G)} ₁(z)+z ^(−1/2) {tilde over(G)} ₀(z)  (33)Equations (32) and (33) are equivalent to:{tilde over (H)}(z ^(1/2))=Det ⁻¹ [−G ₀(z)+z ^(−1/2) G ₁(z)],  (34){tilde over (G)}(z ^(1/2))=Det ⁻¹ [H ₀(z)−z ^(−1/2) H ₁(z)],  (35)due to the relations:{tilde over (H)} ₀(z)=Det ⁻¹ ·G ₁(z),{tilde over (H)} ₁(z)=−Det ⁻¹ ·G₀(z),{tilde over (G)} ₀(z)=−Det ⁻¹ ·H ₁(z),{tilde over (G)}₁(Z)=Det ⁻¹ ·H₀(z),  (36),between the polyphase components of the decomposition and reconstructionfilters in a filter-bank. These relationships can be immediatelyverified by validating the perfect reconstruction condition. In thefollowing, we always assume that the filters H and G are properlyshifted so that Det⁻¹=−1, in order to simplify the expressions. showssuch an example for a biorthogonal point-symmetric filter-pair, namelythe 9/7 transform of Cohen, Daubechics, Feauveau.

By replacing equations (31), (34), (35) in equation (30) we obtain:

$\begin{matrix}{{{A_{x}^{k}\left( z^{\frac{1}{2}} \right)} = {{{z^{\frac{1}{2}}\left( {F_{{4p},0}^{k} + {z^{- \frac{1}{2}}F_{{4p},1}^{k}}} \right)}\left( {G_{0} - {z^{- \frac{1}{2}}G_{1}}} \right)A_{0}^{k + 1}} + {{z^{\frac{1}{2}}\left( {F_{{4p},0}^{k} + {z^{- \frac{1}{2}}F_{{4p},1}^{k}}} \right)}\left( {{- H_{0}} + {z^{- \frac{1}{2}}H_{1}}} \right)D_{0}^{k + 1}} + {T\left( {k,z^{\frac{1}{2}}} \right)}}},} & (37)\end{matrix}$where

$\begin{matrix}{{{T\left( {k,z^{\frac{1}{2}}} \right)} = {{{F_{{4p} + 1}^{k}\left( z^{\frac{1}{2}} \right)}{D_{0}^{k}\left( z^{\frac{1}{2}} \right)}} + {T\left( {{k - 1},z^{\frac{1}{2}}} \right)}}},} & (38)\end{matrix}$is the “‘tail’” that includes the contribution of level k. By groupingtogether the factors of the polyphase components of

F_(4p)^(k)in equation (37) we obtain:

$\begin{matrix}{{A_{x}^{k}\left( z^{\frac{1}{2}} \right)} = {{\left\lbrack {{\left( {{z^{\frac{1}{2}}G_{0}} - G_{1}} \right)F_{{4p},0}^{k}} + {\left( {G_{0} - {z^{- \frac{1}{2}}G_{1}}} \right)F_{{4p},1}^{k}}} \right\rbrack A_{0}^{k + 1}} + {\quad{\quad{\quad\left\lbrack {{\left. \quad{{\left( {H_{1} - {z^{\frac{1}{2}}H_{0}}} \right)F_{{4p},0}^{k}} + {\left( {{z^{- \frac{1}{2}}H_{1}} - H_{0}} \right)F_{{4p},1}^{k}}} \right\rbrack D_{0}^{k + 1}} + {T\left( {k,z^{\frac{1}{2}}} \right)}} \right.}}}}} & (39)\end{matrix}$We denote the factors of the two subbands

A₀^(k + 1), D₀^(k + 1)of equation (39) as:

$\begin{matrix}{{L_{k + 1}^{+ A} = {{\left( {{z^{\frac{1}{2}}G_{0}} - G_{1}} \right)F_{{4p},0}^{k}} + {\left( {G_{0} - {z^{- \frac{1}{2}}G_{1}}} \right)F_{{4p},1}^{k}}}},} & (40) \\{L_{k + 1}^{+ D} = {{\left( {H_{1} - {z^{\frac{1}{2}}H_{0}}} \right)F_{{4p},0}^{k}} + {\left( {{z^{- \frac{1}{2}}H_{1}} - H_{0}} \right){F_{{4p},1}^{k}.}}}} & (41)\end{matrix}$By using these notations, equation (39) is equivalent to:

$\begin{matrix}{{A_{x}^{k}\left( z^{\frac{1}{2}} \right)} = {{L_{k + 1}^{+ A}A_{0}^{k + 1}} + {L_{k + 1}^{+ D}D_{0}^{k + 1}} + {{T\left( {k,z^{\frac{1}{2}}} \right)}.}}} & (42)\end{matrix}$Similarly as above, we can calculate A_(x) ^(k)(−z^(1/2)) by replacing zwith (−z^(1/2)) in equation (27):

$\begin{matrix}{{A_{x}^{k}\left( {- z^{\frac{1}{2}}} \right)} = {{{- z^{\frac{1}{2}}}{F_{4p}^{k}\left( {- z^{\frac{1}{2}}} \right)}{\overset{\sim}{H}\left( {- z^{\frac{1}{2}}} \right)}{A_{0}^{k + 1}(z)}} - {z^{\frac{1}{2}}{F_{4p}^{k}\left( {- z^{\frac{1}{2}}} \right)}{\overset{\sim}{G}\left( {- z^{\frac{1}{2}}} \right)}{D_{0}^{k + 1}(z)}} + {{F_{{4p} + 1}^{k}\left( {- z^{\frac{1}{2}}} \right)}{D_{0}^{k}\left( {- z^{\frac{1}{2}}} \right)}} + {T\left( {{k - 1},{- z^{\frac{1}{2}}}} \right)}}} & (43)\end{matrix}$The filter

$F_{4p}^{k}\left( {- z^{\frac{1}{2}}} \right)$can be written using the Type I polyphase representation as:

$\begin{matrix}{{{F_{4p}^{k}\left( {- z^{\frac{1}{2}}} \right)} = {{F_{{4p},0}^{k}(z)} - {z^{- \frac{1}{2}}{F_{{4p},1}^{k}(z)}}}},} & (44)\end{matrix}$while for point-symmetric, biorthogonal filter-pairs, {tilde over(H)}(−z^(1/2)) and {tilde over (G)}(−z^(1/2)) can be substituted by theType II polyphase representation as:{tilde over (H)}(−z ^(1/2))={tilde over (H)} ₁(z)−z ^(−1/2) {tilde over(H)} ₀(z),  (45){tilde over (G)}(−z ^(1/2))={tilde over (G)} ₁(z)−z ^(−1/2) {tilde over(G)} ₀(z),  (46)equivalent to:{tilde over (H)}(−z ^(1/2))=D ⁻¹ [−G ₀(z)−z ^(−1/2) G ₁(z)],  (47){tilde over (G)}(−z ^(1/2))=D ¹ [H ₀(z)+z ^(−1/2) H ₁(z)],  (48)due to the relations mentioned before—equation (36)—for the polyphasecomponents of the decomposition and reconstruction filters in thefilter-bank.By replacing equations (44), (47) and (48) in equation (43) withDet⁻¹=−1 we get:

$\begin{matrix}{{A_{x}^{k}\left( {- z^{\frac{1}{2}}} \right)} = {{{- {z^{\frac{1}{2}}\left( {F_{{4p},0}^{k} - {z^{- \frac{1}{2}}F_{{4p},1}^{k}}} \right)}}\left( {G_{0} + {z^{- \frac{1}{2}}G_{1}}} \right)A_{0}^{k + 1}} - {{z^{\frac{1}{2}}\left( {F_{{4p},0}^{k} - {z^{- \frac{1}{2}}F_{{4p},1}^{k}}} \right)}\left( {{- H_{0}} - {z^{- \frac{1}{2}}H_{1}}} \right)D_{0}^{k + 1}} + {T\left( {k,{- z^{\frac{1}{2}}}} \right)}}} & (49)\end{matrix}$where T(k,−z^(1/2))=F_(4p+1) ^(k)(−z^(1/2))D₀^(k)(−z^(1/2))+T(k−1,−z^(1/2)) is the “tail” that includes thecontribution of level k. By grouping together the factors of thepolyphase components of

F_(4p)^(k)in equation (49) we get:

$\begin{matrix}{{\left. {{A_{x}^{k}\left( {- z^{\frac{1}{2}}} \right)} = \left\lbrack {{\left( {{{- z^{\frac{1}{2}}}G_{0}} - G_{1}} \right)F_{{4p},0}^{k}} + {\left( {G_{0} + {z^{- \frac{1}{2}}G_{1}}} \right)F_{{4p},1}^{k}\quad}} \right.} \right\rbrack A_{0}^{k + 1}} + {\quad{\quad{{\left. \quad{\left\lbrack {\left( {H_{1} + {z^{\frac{1}{2}}H_{0}}} \right)F_{{4p},0}^{k}} \right\rbrack + {\left( {{{- z^{- \frac{1}{2}}}H_{1}} - H_{0}} \right)F_{{4p},1}^{k}}} \right\rbrack D_{0}^{k + 1}} + {T\left( {k,{- z^{\frac{1}{2}}}} \right)}}}}} & (50)\end{matrix}$We denote the factors of the two subbands

A₀^(k + 1), D₀^(k + 1)as:

$\begin{matrix}{{L_{k + 1}^{- A} = {{\left( {{{- z^{1/2}}G_{0}} - G_{1}} \right)F_{{4p},0}^{k}} + {\left( {G_{0} + {z^{{- 1}/2}G_{1}}} \right)F_{{4p},1}^{k}}}},} & (51) \\{L_{k + 1}^{- D} = {{\left( {H_{1} + {z^{1/2}H_{0}}} \right)F_{{4p},0}^{k}} + {\left( {{{- z^{{- 1}/2}}H_{1}} - H_{0}} \right){F_{{4p},1}^{k}.}}}} & (52)\end{matrix}$By using these notations, equation (50) is equivalent to:

$\begin{matrix}{{A_{x}^{k}\left( {- z^{1/2}} \right)} = {{L_{k + 1}^{- A}A_{0}^{k + 1}} + {L_{k + 1}^{- D}D_{0}^{k + 1}} + {{T\left( {k,{- z^{1/2}}} \right)}.}}} & (53)\end{matrix}$The final expressions of

A_(2x)^(k + 1), D_(2x)^(k + 1)are obtained by replacing equations (42) and (53) in equation (29), asshown in the equation below:

$\begin{matrix}{\begin{bmatrix}A_{2x}^{k + 1} \\D_{2x}^{k + 1}\end{bmatrix} = {{\frac{1}{2}\begin{bmatrix}{{H_{0}(z)} + {z^{{- 1}/2}{H_{1}(z)}}} & {{H_{0}(z)} - {z^{{- 1}/2}{H_{1}(z)}}} \\{{G_{0}(z)} + {z^{{- 1}/2}{G_{1}(z)}}} & {{G_{0}(z)} - {z^{{- 1}/2}{G_{1}(z)}}}\end{bmatrix}}{\quad{\begin{bmatrix}{{L_{k + 1}^{+ A}A_{0}^{k + 1}} + {L_{k + 1}^{+ D}D_{0}^{k + 1}} + {T\left( {k,z^{1/2}} \right)}} \\{{L_{k + 1}^{- A}A_{0}^{k + 1}} + {L_{k + 1}^{- D}D_{0}^{k + 1}} + {T\left( {k,{- z^{1/2}}} \right)}}\end{bmatrix}.}}}} & (54)\end{matrix}$Equation (54) shows that the calculation of the subbands

A_(2x)^(k + 1), D_(2x)^(k + 1)consists of separate calculations of factors like

1/2[(H₀ + z^(−1/2)H₁)L_(k + 1)^(+A) + (H₀ − z^(−1/2)H₁)L_(k + 1)^(−A)]and

1/2[(H₀ + z^(−1/2)H₁)L_(k + 1)^(+D) + (H₀ − z^(−1/2)H₁)L_(k + 1)^(−D)],multiplying the subbands

A₀^(k + 1)  and  D₀^(k + 1)respectively. These factors correspond actually to the predictionfilters of the even-numbered subbands of level k+1. The calculation ofthese factors is done in the following.Part 0: Replacement of

L_(k + 1)^(+A)  and  L_(k + 1)^(−A)in equation (54) for the calculation of

A_(2x)^(k + 1):

$\begin{matrix}{{2F_{8p}^{k + 1}} = {\left. {{\left( {H_{0} + {z^{{- 1}/2}H_{1}}} \right)\left\lbrack {{\left( {{z^{1/2}G_{0}} - G_{1}} \right)F_{{4p},0}^{k}} + {\left( {G_{0} - {z^{1/2}G_{1}}} \right)F_{{4p},1}^{k}}} \right\rbrack} + {\left( {H_{0} - {z^{{- 1}/2}H_{1}}} \right)\left\lbrack {\left( {{{- z^{1/2}}G_{0}} - G_{1}} \right){F_{{4p},0}^{k}++}\left( {G_{0} + {z^{{- 1}/2}G_{1}}} \right)F_{{4p},1}^{k}} \right\rbrack}}\Rightarrow F_{8p}^{k + 1} \right. = {F_{{4p},0}^{k} - {z^{- 1}F_{3}^{1}F_{{4p},1.}^{k}}}}} & (55)\end{matrix}$

-   Part 1: Replacement of

L_(k + 1)^(+D)  and  L_(k + 1)^(−D)

-   in equation (54) for the calculation of

A_(2x)^(k + 1):

$\begin{matrix}{{2F_{{8p} + 1}^{k + 1}} = {\quad{\left( {H_{0} + {z^{{- 1}/2}H_{1}}} \right){\quad{\left\lbrack {{\left( {H_{1} - {z^{1/2}H_{0}}} \right)F_{{4p},0}^{k}} + {\left( {{z^{{- 1}/2}H_{1}} - H_{0}} \right)F_{{4p},1}^{k}}} \right\rbrack + {\left( {H_{0} - {z^{{- 1}/2}H_{1}}} \right){\quad{\left. \left\lbrack {\left( {H_{1} + {z^{1/2}H_{0}}} \right){F_{{4p},0}^{k}++}\left( {{{- z^{{- 1}/2}}H_{1}} - H_{0}} \right)F_{{4p},1}^{k}} \right\rbrack\Rightarrow F_{{8p} + 1}^{k + 1} \right. = {z^{- 1}F_{l}^{1}{F_{{4p},1}^{k}.}}}}}}}}}} & (56)\end{matrix}$

-   Part 2: Replacement of

L_(k + 1)^(+A)  and  L_(k + 1)^(−A)in equation (54) for the calculation of

D_(2x)^(k + 1):

$\begin{matrix}{{2F_{{8p} + 2}^{k + 1}} = {\left. {{\left( {G_{0} + {z^{{- 1}/2}G_{1}}} \right)\left\lbrack {{\left( {{z^{1/2}G_{0}} - G_{1}} \right)F_{{4p},0}^{k}} + {\left( {G_{0} - {z^{{- 1}/2}G_{1}}} \right)F_{{4p},1}^{k}}} \right\rbrack} + {\left( {G_{0} - {z^{{- 1}/2}G_{1}}} \right)\left\lbrack {\left( {{{- z^{1/2}}G_{0}} - G_{1}} \right){F_{{4p},0}^{k}++}\left( {G_{0} + {z^{{- 1}/2}G_{1}}} \right)F_{0,1}^{k}} \right\rbrack}}\Rightarrow F_{{8p} + 2}^{k + 1} \right. = {z^{- 1}F_{2}^{1}{F_{{4p},1}^{k}.}}}} & (57)\end{matrix}$

-   Part 3: Replacement of

L_(k + 1)^(+D)  and  L_(k + 1)^(−D)in equation (54) for the calculation of

D_(2x)^(k + 1):

$\begin{matrix}{{2F_{{8p} + 3}^{k + 1}} = {\left. {{\left( {G_{0} + {z^{{- 1}/2}G_{1}}} \right)\left\lbrack {{\left( {H_{1} - {z^{{- 1}/2}H_{0}}} \right)F_{{4p},0}^{k}} + {\left( {{z^{{- 1}/2}H_{1}} - H_{0}} \right)F_{{4p},1}^{k}}} \right\rbrack} + {\left( {G_{0} - {z^{{- 1}/2}G_{1}}} \right)\left\lbrack {\left( {H_{1} + {z^{1/2}H_{0}}} \right){F_{{4p},0}^{k}++}\left( {{{- z^{{- 1}/2}}H_{1}} - H_{0}} \right) F_{4p{.1}}^{k}} \right\rbrack}}\Rightarrow F_{{8p} + 3}^{k + 1} \right. = {F_{{4p},0}^{k} + {z^{- 1}F_{3}^{1}{F_{{4p},1}^{k}.}}}}} & (58)\end{matrix}$By definition p=0,1, . . . , 2^(k−1)−1.We can complete now equation (54), and hence the propositions for theeven-numbered subbands of the “right” side of the pyramid, atdecomposition level k+1:and with:

$\begin{matrix}{A_{2x}^{k + 1} = {\left. {{F_{8p}^{k + 1}A_{0}^{k + 1}} + {F_{{8p} + 1}^{k + 1}D_{0}^{k + 1}} + {\frac{1}{2}\left\lbrack {{\left( {H_{0} + {z^{- \frac{1}{2}}H_{1}}} \right){T\left( {k,z^{\frac{1}{2}}} \right)}} + {\left( {H_{0} - {z^{- \frac{1}{2}}H_{1}}} \right){T\left( {k,{- z^{\frac{1}{2}}}} \right)}}} \right\rbrack}}\Rightarrow\Rightarrow A_{2x}^{k + 1} \right. = {{F_{8p}^{k + 1}A_{0}^{k + 1}} + {F_{{8p} + 1}^{k + 1}D_{0}^{k + 1}} + \left\lbrack {{H(z)}{T\left( {k,z} \right)}} \right\rbrack_{0}}}} & (59) \\{D_{2x}^{k + 1} = {\left. {{F_{{8p} + 2}^{k + 1}A_{0}^{k + 1}} + {F_{{8p} + 3}^{k + 1}D_{0}^{k + 1}} + {\frac{1}{2}\left\lbrack {{\left( {G_{0} + {z^{- \frac{1}{2}}G_{1}}} \right){T\left( {k,z^{\frac{1}{2}}} \right)}} + {\left( {G_{0} - {z^{- \frac{1}{2}}G_{1}}} \right){T\left( {k,{- z^{\frac{1}{2}}}} \right)}}} \right\rbrack}}\Rightarrow\Rightarrow D_{2x}^{k + 1} \right. = {{F_{{8p} + 2}^{k + 1}A_{0}^{k + 1}} + {F_{{8p} + 3}^{k + 1}D_{0}^{k + 1}} + \left\lbrack {{G(z)}{T\left( {k,z} \right)}} \right\rbrack_{0}}}} & (60) \\{{T\left( {k,z} \right)} = {{F_{{4p} + 1}^{k}D_{0}^{k}} + {\left( {{H \cdot F_{i{(1)}}^{k - 1}}D_{0}^{k - 1}} \right)b_{0}} + \left\lbrack {H\left( {{H \cdot F_{i{(2)}}^{k - 2}}D_{0}^{k - 2}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + {\left\lbrack {H\left( {{H\left( {\ldots\left( {{H \cdot F_{i{({k - 1})}}^{1}}D_{0}^{1}} \right)}_{b_{k - 2}} \right)}_{b_{k - 3}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}.}}} & (61)\end{matrix}$Define y=2x. Since xε[2^(k−1),2^(k)−1] it follows thatyε[2^(k),2^(k+1)−2]. The equivalent of equation (7) is:y=2^(k) +q,  (62)with qε[0,2^(k)−2]. Also, the definition of y and equations (7) and (62)imply that:q=2p.  (63)This equation shows that q is an even number, therefore its binaryrepresentation is given by

${q = {\sum\limits_{i = 0}^{k - 1}{c_{i} \cdot 2^{i}}}},{p = {\sum\limits_{i = 0}^{k - 2}{b_{i} \cdot 2^{i}}}},$with c_(i)ε{0,1}, ∀iε[1,k−1] and c₀=0. The binary representation of p istherefore by using (63) one can write:

$\begin{matrix}{{\sum\limits_{i = 1}^{k - 1}{c_{i} \cdot 2^{i}}} = {\sum\limits_{i = 0}^{k - 2}{b_{1} \cdot {2^{i + 1}.}}}} & (64)\end{matrix}$From (64) it follows that c_(i)=b_(i−1) for any iε[1,k−1].By using these observations and by replacing the expression of the“tail” T given in equation (61), one can write equations (59) and (60)as follows:

$\begin{matrix}{{A_{y}^{k + 1} = {{F_{4q}^{k + 1}A_{0}^{k + 1}} + {F_{{4q} + 1}^{k + 1}D_{0}^{k + 1}} + \left\lbrack {{H \cdot F_{{4{\lfloor\frac{q}{2}\rfloor}} + 1}^{k}}D_{0}^{k}} \right\rbrack_{c_{0}} + \left\lbrack {H\left( {{H \cdot F_{j{(2)}}^{k - 1}}D_{0}^{k - 1}} \right)}_{c_{1}} \right\rbrack_{c_{0}} + \left\lbrack {H\left( {H\left( {{H \cdot F_{j{(3)}}^{k - 2}}D_{0}^{k - 2}} \right)}_{c_{2}} \right)}_{c_{1}} \right\rbrack_{c_{0}} + {{\ldots++}\left\lbrack {H\left( {{H\left( {\ldots\left( {{H \cdot F_{j{(k)}}^{1}}D_{0}^{1}} \right)}_{c_{k - 1}} \right)}_{c_{k - 2}}\ldots} \right)}_{c_{1}} \right\rbrack}_{c_{0}}}},} & (65)\end{matrix}$and

$\begin{matrix}{{D_{y}^{k + 1} = {{F_{{4q} + 2}^{k + 1}A_{0}^{k + 1}} + {F_{{4q} + 3}^{k + 1}D_{0}^{k + 1}} + \left\lbrack {{G \cdot F_{{4{\lfloor\frac{q}{2}\rfloor}} + 1}^{k}}D_{0}^{k}} \right\rbrack_{c_{0}} + \left\lbrack {G\left( {{H \cdot F_{j{(2)}}^{k - 1}}D_{0}^{k - 1}} \right)}_{c_{1}} \right\rbrack_{c_{0}} + \left\lbrack {G\left( {H\left( {{H \cdot F_{j{(3)}}^{k - 2}}D_{0}^{k - 2}} \right)}_{c_{2}} \right)}_{c_{1}} \right\rbrack_{c_{0}} + {{\ldots++}\left\lbrack {G\left( {{H\left( {\ldots\left( {{H \cdot F_{j{(k)}}^{1}}D_{0}^{1}} \right)}_{c_{k - 1}} \right)}_{c_{k - 2}}\ldots} \right)}_{c_{1}} \right\rbrack}_{c_{0}}}},} & (66)\end{matrix}$where j(m) are defined as

${{j(m)} = {{4\left\lfloor \frac{q}{2^{m}} \right\rfloor} + 1}},$for any mε[1,k].Equations (65), (66) are equivalent to equations (25) for the evenvalues of x (even-numbered subbands). Hence, in this case thepropositions P_(R)(k+1) are true.The proof of P_(R)(k+1) for odd values of x is done in a similar manner.In order to calculate the subbands

A_(2x + 1)^(k + 1), D_(2x + 1)^(k + 1),we perform a single-level forward transform retaining the odd samples(“complementary” transform):

$\begin{matrix}{\begin{bmatrix}A_{{2x} + 1}^{k + 1} \\D_{{2x} + 1}^{k + 1}\end{bmatrix} = {{{\frac{1}{2}\begin{bmatrix}{{H_{0}(z)} + {z^{\frac{1}{2}}{H_{1}(z)}}} & {{H_{1}(z)} - {z^{\frac{1}{2}}{H_{0}(z)}}} \\{{G_{0}(z)} + {z^{\frac{1}{2}}{G_{1}(z)}}} & {{G_{1}(z)} - {z^{\frac{1}{2}}{G_{0}(z)}}}\end{bmatrix}}\begin{bmatrix}{A_{x}^{k}\left( z^{\frac{1}{2}} \right)} \\{A_{x}^{k}\left( {- z^{\frac{1}{2}}} \right)}\end{bmatrix}}.}} & (67)\end{matrix}$Notice that the terms A_(x) ^(k)(z^(1/2)),A_(z) ^(k)(−z^(1/2)) arealready calculated in equations (42) and (53). Replacing theirexpressions in equation (67) yields:

$\begin{matrix}\begin{matrix}{\begin{bmatrix}A_{2x}^{k + 1} \\D_{2x}^{k + 1}\end{bmatrix} = {\frac{1}{2}\begin{bmatrix}{{H_{0}(z)} + {z^{\frac{1}{2}}{H_{1}(z)}}} & {{H_{1}(z)} - {z^{\frac{1}{2}}{H_{0}(z)}}} \\{{G_{0}(z)} + {z^{\frac{1}{2}}{G_{1}(z)}}} & {{G_{1}(z)} - {z^{\frac{1}{2}}{G_{0}(z)}}}\end{bmatrix}}} \\{\begin{bmatrix}{{L_{k + 1}^{+ A}A_{0}^{k + 1}} + {L_{k + 1}^{+ D}D_{0}^{k + 1}} + {T\left( {k,z^{\frac{1}{2}}} \right)}} \\{{L_{k + 1}^{- A}A_{0}^{k + 1}} + {L_{k + 1}^{- D}D_{0}^{k + 1}} + {T\left( {k,{- z^{\frac{1}{2}}}} \right)}}\end{bmatrix}.}\end{matrix} & (68)\end{matrix}$Equation (68) shows that the calculation of the subbands

A_(2x + 1)^(k + 1), D_(2x + 1)^(k + 1)consists of separate calculations of factors like

$\frac{1}{2}\left\lbrack {{\left( {H_{0} + {z^{\frac{1}{2}}H_{1}}} \right)L_{k + 1}^{+ A}} + {\left( {H_{1} - {z^{\frac{1}{2}}H_{0}}} \right)L_{k + 1}^{- A}}} \right\rbrack$and

${\frac{1}{2}\left\lbrack {{\left( {H_{0} + {z^{\frac{1}{2}}H_{1}}} \right)L_{k + 1}^{+ D}} + {\left( {H_{1} - {z^{\frac{1}{2}}H_{0}}} \right)L_{k + 1}^{- D}}} \right\rbrack},$multiplying the subbands

A₀^(k + 1) b  and  D₀^(k + 1)respectively. These factors correspond actually to the predictionfilters of the odd-numbered subbands of level k+1. The calculation ofthese factors is done in the following.

-   Part 4: Replacement of

L_(k + 1)^(+A)  and  L_(k + 1)^(−A)in equation (68) for the calculation of

A_(2x + 1)^(k + 1):

$\begin{matrix}{{2F_{{8p} + 4}^{k + 1}} = {\left. {{\left( {H_{1} + {z^{\frac{1}{2}}H_{0}}} \right)\left\lbrack {{\left( {{z^{\frac{1}{2}}G_{0}} - G_{1}} \right)F_{{4p},0}^{k}} + {\left( {G_{0} - {z^{- \frac{1}{2}}G_{1}}} \right)F_{{4p},1}^{k}}} \right\rbrack} + {\left( {H_{1} - {z^{\frac{1}{2}}H_{0}}} \right)\left\lbrack {\left( {{{- z^{\frac{1}{2}}}G_{0}} - G_{1}} \right){F_{{4p},0}^{k}++}\left( {G_{0} + {z^{- \frac{1}{2}}G_{1}}} \right)F_{{4p},1}^{k}} \right\rbrack}}\Rightarrow F_{{8p} + 4}^{k + 1} \right. = {F_{{4p},1}^{k} + {F_{0}^{1}{F_{{4p},0}^{k}.}}}}} & (69)\end{matrix}$

-   Part 5: Replacement of

L_(k + 1)^(+D)  and  L_(k + 1)^(−D)in equation (68) for the calculation of

A_(2x + 1)^(k + 1):

$\begin{matrix}{{2F_{{8p} + 5}^{k + 1}} = {\left. {{\left( {H_{1} + {z^{\frac{1}{2}}H_{0}}} \right)\left\lbrack {{\left( {H_{1} - {z^{\frac{1}{2}}H_{0}}} \right)F_{{4p},0}^{k}} + {\left( {{z^{- \frac{1}{2}}H_{1}} - H_{0}} \right)F_{{4p},1}^{k}}} \right\rbrack} + {\left( {H_{1} - {z^{\frac{1}{2}}H_{0}}} \right)\left\lbrack {\left( {H_{1} + {z^{\frac{1}{2}}H_{0}}} \right){F_{{4p},0}^{k}++}\left( {{{- z^{- \frac{1}{2}}}H_{1}} - H_{0}} \right)F_{{4p},1}^{k}} \right\rbrack}}\Rightarrow F_{{8p} + 5}^{k + 1} \right. = {F_{1}^{1}{F_{{4p},0}^{k}.}}}} & (70)\end{matrix}$

-   Part 6: Replacement of

L_(k + 1)^(+A)  and  L_(k + 1)^(−A)in equation (68) for the calculation of

D_(2x + 1)^(k + 1):

$\begin{matrix}{{2F_{{8p} + 6}^{k + 1}} = {\left. {{\left( {G_{1} + {z^{\frac{1}{2}}G_{0}}} \right)\left\lbrack {{\left( {{z^{\frac{1}{2}}G_{0}} - G_{1}} \right)F_{{4p},0}^{k}} + {\left( {G_{0} - {z^{- \frac{1}{2}}G_{1}}} \right)F_{{4p},1}^{k}}} \right\rbrack} + {\left( {G_{1} - {z^{\frac{1}{2}}G_{0}}} \right)\left\lbrack {\left( {{{- z^{\frac{1}{2}}}G_{0}} - G_{1}} \right){F_{{4p},0}^{k}++}\left( {G_{0} + {z^{- \frac{1}{2}}G_{1}}} \right)F_{{4p},1}^{k}} \right\rbrack}}\Rightarrow F_{{8p} + 6}^{k + 1} \right. = {F_{2}^{1}{F_{{4p},0}^{k}.}}}} & (71)\end{matrix}$

-   Part 7: Replacement of

L_(k + 1)^(+D)  and  L_(k + 1)^(−D)in equation (68) for the calculation of

D_(2x + 1)^(k + 1):

$\begin{matrix}{{2F_{{8p} + 7}^{k + 1}} = {\left. {{\left( {G_{1} + {z^{\frac{1}{2}}G_{0}}} \right)\left\lbrack {{\left( {H_{1} - {z^{\frac{1}{2}}H_{0}}} \right)F_{{4p},0}^{k}} + {\left( {{z^{- \frac{1}{2}}H_{1}} - H_{0}} \right)F_{{4p},1}^{k}}} \right\rbrack} + {\left( {G_{1} - {z^{\frac{1}{2}}G_{0}}} \right)\left\lbrack {\left( {H_{1} - {z^{\frac{1}{2}}H_{0}}} \right){F_{{4p},0}^{k}++}\left( {{{- z^{- \frac{1}{2}}}H_{1}} - H_{0}} \right)F_{{4p},1}^{k}} \right\rbrack}}\Rightarrow F_{{8p} + 7}^{k + 1} \right. = {F_{{4p},1}^{k} + {F_{3}^{1}{F_{{4p},0}^{k}.}}}}} & (72)\end{matrix}$By definition p=0,1, . . . , 2^(k−1)−1.We can complete now equation (68), and hence the second proposition oflevel k+1:

$\begin{matrix}{{A_{{2x} + 1}^{k + 1} = {\left. {{F_{{8p} + 4}^{k + 1}A_{0}^{k + 1}} + {F_{{8p} + 5}^{k + 1}D_{0}^{k + 1}} + {\frac{1}{2}\left\lbrack {{\left( {H_{1} + {z^{\frac{1}{2}}H_{0}}} \right){T\left( {k,z^{\frac{1}{2}}}\; \right)}} + {\left( {H_{1} - {z^{\frac{1}{2}}\; H_{0}}} \right){T\left( {k,{- z^{\frac{1}{2}}}} \right)}}} \right\rbrack}}\Rightarrow\Rightarrow A_{{2x} + 1}^{k + 1} \right. = {{F_{{8p} + 4}^{k + 1}A_{0}^{k + 1}} + {F_{{8p} + 5}^{k + 1}D_{0}^{k + 1}} + \left\lbrack {{H(z)}{T\left( {k,z} \right)}} \right\rbrack_{1}}}},{and}} & (73) \\{D_{{2x} + 1}^{k + 1} = {\left. {{F_{{8p} + 6}^{k + 1}A_{0}^{k + 1}} + {F_{{8p} + 7}^{k + 1}D_{0}^{k + 1}} + {\frac{1}{2}\left\lbrack {{\left( {G_{1} + {z^{\frac{1}{2}}G_{0}}} \right){T\left( {k,z^{\frac{1}{2}}} \right)}} + {\left( {G_{1} - {z^{\frac{1}{2}}G_{0}}} \right){T\left( {k,{- z^{\frac{1}{2}}}} \right)}}} \right\rbrack}}\Rightarrow\Rightarrow D_{{2x} + 1}^{k + 1} \right. = {{F_{{8p} + 6}^{k + 1}A_{0}^{k + 1}} + {F_{{8p} + 7}^{k + 1}D_{0}^{k + 1}} + {\left\lbrack {{G(z)}{T\left( {k,z} \right)}} \right\rbrack_{1}{with}}}}} & (74) \\{{T\left( {k,z} \right)} = {{F_{{4p} + 1}^{k}D_{0}^{k}} + \left( {{H \cdot F_{i{(1)}}^{k - 1}}D_{0}^{k - 1}} \right)_{b_{0}} + \left\lbrack {H\left( {{H \cdot F_{i{(2)}}^{k - 2}}D_{0}^{k - 2}} \right)}_{b_{1}} \right\rbrack_{b_{0}} + \ldots + {\left\lbrack {H\left( {{H\left( {{\ldots\left( {{H \cdot F_{i{({k - 1})}}^{1}}D_{0}^{1}} \right)}b_{k - 2}} \right)}_{b_{k - 3}}\ldots} \right)}_{b_{1}} \right\rbrack_{b_{0}}.}}} & (75)\end{matrix}$Define y=2x+1. Since xε[2^(k−1),2^(k)−1] it follows thatyε[2^(k),2^(k+1)−1]. The equivalent of equation (7) is:y=2^(k) +q,  (76)with qε[0,2^(k)−1]. Also, the definition of y and equations (7) and (76)imply that:q=2p+1.  (77)This equation shows that q is an odd number, therefore its binaryrepresentation is given by

$\begin{matrix}{{q = {\underset{i = 0}{\sum\limits^{k - 1}}{c_{i} \cdot 2^{i}}}},} \\{{p = {\underset{i = 0}{\sum\limits^{k - 2}}{b_{i} \cdot 2^{i}}}},}\end{matrix}$with c_(i)ε{0,1}, ∀iε[1,k−1] and c₀=1. The binary representation of p istherefore by using (77) one can write:

$\begin{matrix}{{\underset{i = 1}{\sum\limits^{k - 1}}{c_{i} \cdot 2^{i}}} = {\underset{i = 0}{\sum\limits^{k - 2}}{b_{i} \cdot {2^{i + 1}.}}}} & (78)\end{matrix}$From (78) again it follows that c_(i)=b_(i−1) for any iε[1,k−1]. Also,from (77) it follows that

$p = {\left\lfloor \frac{q}{2} \right\rfloor.}$By using these observations and by replacing the expression of the“tail” T, given in equation (75), one can write equations (73) and (74)as follows:

$\begin{matrix}{A_{y}^{k + 1} = {{F_{4q}^{k + 1}A_{0}^{k + 1}F_{{4q} + 1}^{k + 1}D_{0}^{k + 1}} + \left( {{H \cdot F_{{4{\lfloor\frac{q}{2}\rfloor}} + 1}^{k}}D_{0}^{k}} \right)_{c_{o}} + \left\lbrack {H\left( {{H \cdot F_{j{(2)}}^{k - 1}}D_{0}^{k - 1}} \right)}_{c_{i}} \right\rbrack_{c_{o}} + \left\lbrack {H\left( {H\left( {{H \cdot F_{j{(3)}}^{k - 2}}D_{0}^{k - 2}} \right)}_{c2} \right)}_{c_{i}} \right\rbrack_{c_{o}} + {¨++}}} & (79) \\{\left. \left\lbrack {H\left( {{H\left( {..\left( {{H \cdot F_{j{(k)}}^{1}}D_{0}^{1}} \right)_{c_{k}}} \right)}_{c_{k} - 1}¨} \right)}_{c_{2}} \right)_{c_{1}} \right\rbrack_{c_{o}},{and}} & \; \\{D_{y}^{k + 1} = {{{{F_{{4q} + 2}^{k + 1}A_{0}^{k + 1}} + {F_{{4q} + 3}^{k + 1}D_{0}^{k + 1}} + \left( {{G \cdot F_{{4{\lfloor\frac{q}{2}\rfloor}} + 1}^{k}}D_{0}^{k}} \right)_{c_{o}} + \left\lbrack {G\left( {{H \cdot F_{j{(2)}}^{k - 1}}D_{0}^{k - 1}} \right)}_{c_{i}} \right\rbrack_{c_{o}} + \left\lbrack {G\left( {H\left( {{H \cdot F_{j{(3)}}^{k - 2}}D_{0}^{k - 2}} \right)}_{c2} \right)}_{c_{i}} \right\rbrack_{c_{o}} +}..} +}} & (80) \\{\left. {+ \left. \left\lbrack {G\left( {{H\left( {¨\left( {{H \cdot F_{j{(k)}}^{1}}D_{0}^{1}} \right)}_{c_{k}} \right)}_{c_{k} - i}¨} \right)}_{c_{2}} \right. \right)_{c_{i}}} \right\rbrack_{c_{o}},} & \;\end{matrix}$where j(m) are defined as

${j(m)} = {{4\left\lfloor \frac{q}{2^{m}} \right\rfloor} + 1}$for any mε[1,k].Equations (79), (80) are equivalent with equations (25) for the oddvalues of x (odd-numbered subbands) Hence the propositions P_(R)(k+1)are true in this case too. By joining equations (65), (66) with (79),(80) we derive the set of propositions P_(R)(k+1) of level k+1, for anyvalues of x. One can also verify immediately that the filters calculatedby equations (55)-(58), (69)-(71) can be derived from the generaldescription of equations (9)-(16) for every iε[0,2^(k−1)−1], if in theseequations k is replaced with k+1. Hence, the derivations of the filtersfor level k+1 are true.This leads to the conclusion that the propositions P(k+1) are true forthe case of E=k+1. This means that they are true for every E, E≧1.Properties of the Prediction Filters for Biorthogonal Point-SymmetricFilter-Pairs

Several symmetry properties of the prediction filters derived in theprevious subsection for an arbitrary decomposition level k aredemonstrated. The general form of the prediction filters for level k isgiven in equations (9)-(16). The prediction filters of level k+1 areobtained from (9)-(16) by replacing k with k+1:

$\begin{matrix}{{F_{8i}^{k + 1} = {F_{{4i},0}^{k} - {z^{- 1}F_{3}^{1}F_{{4i},1}^{k}}}},} & (81) \\{{F_{{8i} + 1}^{k + 1} = {z^{- 1}F_{1}^{1}F_{{4i},1}^{k}}},} & (82) \\{{F_{{8i} + 2}^{k + 1} = {z^{- 1}F_{2}^{1}F_{{4i},1}^{k}}},} & (83) \\{{F_{{8i} + 3}^{k + 1} = {F_{{4i},0}^{k} + {z^{- 1}F_{3}^{1}F_{{4i},1}^{k}}}},} & (84) \\{{F_{{8i} + 4}^{k + 1} = {F_{{4i},1}^{k}F_{0}^{1}F_{{4i},0}^{k}}},} & (85) \\{{F_{{8i} + 5}^{k + 1} = {F_{1}^{1}F_{{4i},0}^{k}}},} & (86) \\{{F_{{8i} + 6}^{k + 1} = {F_{2}^{1}F_{{4i},0}^{k}}},} & (87) \\{{F_{{8i} + 7}^{k + 1} = {F_{{4i},1}^{k} + {F_{3}^{1}F_{{4i},0}^{k}}}},} & (88)\end{matrix}$where i=0,1, . . . , 2^(k−1)−1.The mathematical formalizm describing the symmetry properties proven inthis section is expressed for biorthogonal point-symmetric filters bythe set of propositions given below:

${P_{s}(k)}:\left\{ \begin{matrix}{{{F_{4m}^{k}(z)} = {{zF}_{4{({2^{k - 1} - m - 1})}}^{k}\left( z^{- 1} \right)}},} & \; & \; & \; & (89) \\{{{F_{{4m} + 1}^{k}(z)} = {F_{{4{({2^{k - 1} - m - 1})}} + 1}^{k}\left( z^{- 1} \right)}},} & \; & \; & \; & (90) \\{{{F_{{4m} + 2}^{k}(z)} = {z^{2}{F_{{4{({2^{k - 1} - m - 1})}} + 2}^{k}\left( z^{- 1} \right)}}},} & \; & \; & \; & (91) \\{{F_{{4m} + 3}^{k}(z)} = {{{zF}_{{4{({2^{k - 1} - m - 1})}} + 3}^{k}\left( z^{- 1} \right)}.}} & \; & \; & \; & (92)\end{matrix} \right.$for m=0,1, . . . , 2^(k−2)−1 and k>1.These equations indicate the fact that we can derive half of theF-filters of level k as the time-inverses of the other half of the setof filters for the same level under by some shifts. Specifically, thefilters are time-inversed in groups of four filters that lay in a“mirror” fashion in the group of the prediction filters. Thus, the firstfour F-filters are related with the last four, the second four F-filterswith the penultimate four, and so on.

These properties will be demonstrated by mathematical induction. Thus wefirst give the proof of P_(S)(2). Then we assume that the equationsP_(S)(k) are true, and based on this assumption we derive the symmetryproperties P_(S)(k+1) between the F-filters of level k+1. If the derivedset of equations P_(S)(k+1) can be simply obtained from P_(S)(k) byreplacing k with k+1, then according to the induction principle thepropositions P_(S)(E) are true for any level E, E>1.

Throughout the proofs given in this paper, the following relationshipsare used for the prediction filters of the first decomposition level:

$\begin{matrix}{{{F_{0}^{1}\left( z^{- 1} \right)} = {z^{- 1}{F_{0}^{1}(z)}}},} & (93) \\{{{F_{1}^{1}\left( z^{- 1} \right)} = {F_{1}^{1}(z)}},} & (94) \\{{{F_{2}^{1}\left( z^{- 1} \right)} = {z^{- 2}{F_{2}^{1}(z)}}},} & (95) \\{{{F_{3}^{1}\left( z^{- 1} \right)} = {z^{- 1}{F_{3}^{1}(z)}}},} & (96) \\{{{F_{0}^{1}(z)} = {- {F_{3}^{1}(z)}}},} & (97) \\{{F_{3}^{1}(z)} = {- {{{zF}_{0}^{1}\left( z^{- 1} \right)}.}}} & (98)\end{matrix}$These properties are proven in the appendix II for biorthogonalpoint-symmetric filters. The symmetry propositions P_(S)(2)corresponding to the case k=2 are:

${P_{s}(2)}:\left\{ \begin{matrix}{{{F_{0}^{2}(z)} = {{zF}_{4}^{2}\left( z^{- 1} \right)}},} & \; & \; & \; & (99) \\{{{F_{1}^{2}(z)} = {F_{5}^{2}\left( z^{- 1} \right)}},} & \; & \; & \; & (100) \\{{{F_{2}^{2}(z)} = {z^{2}{F_{6}^{2}\left( z^{- 1} \right)}}},} & \; & \; & \; & (101) \\{{F_{3}^{2}(z)} = {{{zF}_{7}^{2}\left( z^{- 1} \right)}.}} & \; & \; & \; & (102)\end{matrix} \right.$The proof of these equations is given in appendix III. We assume thatthe symmetry propositions P_(S)(k) are true and based on them we derivethe symmetry relationships for level k+1. Consider equation (89), whichis true by assumption, and replace z with z^(1/2) and −z^(1/2)respectively. The following two equations are obtained:

$\begin{matrix}{{{F_{4m}^{k}\left( z^{1/2} \right)} = {z^{1/2}{F_{4{({2^{k - 1} - m - 1})}}^{k}\left( z^{{- 1}/2} \right)}}},} & (103) \\{{F_{4m}^{k}\left( {- z^{1/2}} \right)} = {{- z^{1/2}}{{F_{4{({2^{k - 1} - m - 1})}}^{k}\left( {- z^{{- 1}/2}} \right)}.}}} & (104)\end{matrix}$The polyphase components of F_(4m) ^(k), with m=0,1, . . . , 2^(k−2)−1are given by:

$\begin{matrix}{{{F_{{4m},0}^{k}(z)} = {1/{2\left\lbrack {{F_{4m}^{k}\left( z^{1/2} \right)} + {F_{4m}^{k}\left( {- z^{1/2}} \right)}} \right\rbrack}}},} & (105) \\{{F_{4m{.1}}^{k}(z)} = {{1/2}{{z^{1/2}\left\lbrack {{F_{4m}^{k}\left( z^{1/2} \right)} - {F_{4m}^{k}\left( {- z^{1/2}} \right)}} \right\rbrack}.}}} & (106)\end{matrix}$In equations (105), (106) we can replace the terms

F_(4m)^(k)(z^(1/2)), F_(4m)^(k)(−z^(1/2))by using (103), (104), yielding:

$\begin{matrix}{{{F_{{4m},0}^{k}(z)} = {\left. {{1/2}{z^{1/2}\left\lbrack {{F_{4{({2^{k - 1} - m - 1})}}^{k}\left( z^{{- 1}/2} \right)} - {F_{4{({2^{k - 1} - m - 1})}}^{k}\left( {- z^{{- 1}/2}} \right)}} \right\rbrack}}\Leftrightarrow{F_{{4m},0}^{k}(z)} \right. = {{zF}_{{4{({2^{k - 1} - m - 1})}},1}^{k}\left( z^{- 1} \right)}}},} & (107) \\{{{F_{{4m},1}^{k}(z)} = {\left. {{1/2}{z\left\lbrack {{F_{4{({2^{k - 1} - m - 1})}}^{k}\left( z^{{- 1}/2} \right)} + {F_{4{({2^{k - 1} - m - 1})}}^{k}\left( {- z^{{- 1}/2}} \right)}} \right\rbrack}}\Leftrightarrow{F_{{4m},1}^{k}(z)} \right. = {{zF}_{{4{({2^{k - 1} - m - 1})}},0}^{k}\left( z^{- 1} \right)}}},} & (108)\end{matrix}$with m=0,1, . . . , 2^(k−2)−1 and k>1.Let us start by proving the symmetry properties for the filters of theform

F_(4m)^(k + 1),with m=0,1, . . . , 2^(k−1)−1; these properties are separately derivedfor m even and m odd. For the case of m even, we define m=2j, with2j=0,2,4, . . . , 2^(k−1)−2, equivalent to j=0,1,2, . . . , 2^(k−2)−1.Hence, from equations (81) and (98) we have:

$\begin{matrix}{{F_{4m}^{k + 1}(z)} = {{F_{8j}^{k + 1}(z)} = {\left. {{F_{{4j},0}^{k}(z)} - {z^{- 1}{F_{3}^{1}(z)}{F_{{4j},1}^{k}(z)}}}\Leftrightarrow{F_{4m}^{k + 1}(z)} \right. = {{F_{{4j},0}^{k}(z)} + {{F_{0}^{1}\left( z^{- 1} \right)}{{F_{{4j},1}^{k}(z)}.}}}}}} & (109)\end{matrix}$Since j=0,1, . . . , 2^(k−2)−1, we can substitute (107) and (108) inequation (109), obtaining that:

$\begin{matrix}{{F_{4m}^{k + 1}(z)} = {{z\left( {{F_{{4{({2^{k - 1} - j - 1})}},1}^{k}\left( z^{- 1} \right)} + {{F_{0}^{1}\left( z^{- 1} \right)} \cdot {F_{{4{({2^{k - 1} - j - 1})}},0}^{k}\left( z^{- 1} \right)}}} \right)}.}} & (110)\end{matrix}$From (85) in which z is replaced with z⁻¹ and from the definition of mwe obtain that:

$\begin{matrix}{{{{F_{4m}^{k + 1}(z)} = {\left. {{zF}_{{8{({2^{k - 1} - j - 1})}} + 4}^{k + 1}\left( z^{- 1} \right)}\Leftrightarrow{F_{4m}^{k + 1}(z)} \right. = {{zF}_{4{({2^{k} - m - 1})}}^{k + 1}\left( z^{- 1} \right)}}},{{{for}\mspace{14mu} m} = 0},2,4,\ldots\mspace{11mu},{2^{k - 1} - 2.}}\mspace{335mu}} & (111)\end{matrix}$For the case of m odd, we may define m=2j+1 with 2j+1=1,3,5, . . .2^(k−1)−1, equivalent to j=0,1,2, . . . , 2^(k−2)−1. Thus, fromequations (85) and (98) we have:

$\begin{matrix}{{F_{4m}^{k + 1}(z)} = {{F_{{8j} + 4}^{k + 1}(z)} = {\left. {{F_{{4j},1}^{k}(z)} + {{F_{0}^{1}(z)}{F_{{4j},0}^{k}(z)}}}\Leftrightarrow{F_{4m}^{k + 1}(z)} \right. = {{F_{{4j},1}^{k}(z)} - {{{zF}_{3}^{1}\left( z^{- 1} \right)}{{F_{{4j},0}^{k}(z)}.}}}}}} & (112)\end{matrix}$Since j=0,1, . . . , 2^(k−2)−1, we can substitute (107) and (108) inequation (112) and we get:

$\begin{matrix}{{F_{4m}^{k + 1}(z)} = {{z\left( {{F_{{4{({2^{k - 1} - j - 1})}},0}^{k}\left( z^{- 1} \right)} - {{{zF}_{3}^{1}\left( z^{- 1} \right)}{F_{{4{({2^{k - 1} - j - 1})}},1}^{k}\left( z^{- 1} \right)}}} \right)}.}} & (113)\end{matrix}$From (81) in which z is replaced with z⁻¹ and from the definition of mwe obtain that:

$\begin{matrix}{{{F_{4m}^{k + 1}(z)} = {\left. {{zF}_{8{({2^{k - 1} - j - 1})}}^{k + 1}\left( z^{- 1} \right)}\Leftrightarrow{F_{4m}^{k + 1}(z)} \right. = {{zF}_{4{({2^{k} - m - 1})}}^{k + 1}\left( z^{- 1} \right)}}},} & (114)\end{matrix}$for m=1,3,5, . . . , 2^(k−1)−1. Thus combining the results of equations(111) and (114), we have:

$\begin{matrix}{{{F_{4m}^{k + 1}(z)} = {{{{zF}_{4{({2^{k} - m - 1})}}^{k + 1}\left( z^{- 1} \right)}\mspace{14mu}{for}\mspace{14mu} m} = 0}},1,2,\ldots\mspace{11mu},{2^{k - 1} - 1.}} & (115)\end{matrix}$To prove the symmetry properties for the prediction filters of the formF_(4m) ^(k+1), with m=0,1, . . . , 2^(k−1)−1, we follow the samerationale as before. Hence, these properties are separately derived form even and m odd. For the case of m even, we denote m=2j with 2j=0,2,4,. . . , 2^(k−1)−2 equivalent to j=0,1,2, . . . , 2^(k−2)−1. By usingequations (82) and (94) we obtain:F _(4m+1) ^(k+1)(z)=F _(8j+1) ^(k+1)(z)=z ⁻¹ F ₁ ¹(z)F _(4j,1) ^(k)(z)

F _(4m+1) ^(k+1)(z)=z ⁻¹ F ₁ ¹(z ⁻¹)F _(4j,1) ^(k)(z).  (116)Since j=0,1, . . . , 2^(k−2)−1, we can substitute (108) in equation(116), obtaining:F _(4m+1) ^(k+1)(z)=F ₁ ¹(z ⁻¹)F ₄₍₂ _(k−1) _(−j−1),0) ^(k)(z ⁻¹⁾From (86) in which z is replaced with z⁻¹ and from the definition of m,we obtain that:

$\begin{matrix}{{{{F_{{4m} + 1}^{k + 1}(z)} = {\left. {F_{{8{({2^{k - 1} - j - 1})}} + 5}^{k + 1}\left( z^{- 1} \right)}\Leftrightarrow{F_{{4m} + 1}^{k + 1}(z)} \right. = {F_{{4{({2^{k} - m - 1})}} + 1}^{k + 1}\left( z^{- 1} \right)}}},{{{for}\mspace{14mu} m} = 0},2,4,\ldots\mspace{11mu},{2^{k - 1} - 2.}}} & (117)\end{matrix}$For the case of m odd, we denote m=2j+1 with 2j+1=1,3,5, . . . ,2^(k−1)−1 equivalent to j=0,1,2, . . . , 2^(k−2)−1. Thus, from equations(87) and (94) we obtain:

$\begin{matrix}{{F_{{4m} + 1}^{k + 1}(z)} = {{F_{{8j} + 5}^{k + 1}(z)} = {\left. {{F_{1}^{1}(z)}{F_{{4j},0}^{k}(z)}}\Leftrightarrow{F_{{4m} + 1}^{k + 1}(z)} \right. = {{F_{1}^{1}\left( z^{- 1} \right)}{{F_{{4j},0}^{k}(z)}.}}}}} & (118)\end{matrix}$Since j=0,1, . . . , 2^(k−2)−1, we can substitute (107) in equation (118), obtaining:

$\begin{matrix}{{F_{{4m} + 1}^{k + 1}(z)} = {{{zF}_{1}^{1}\left( z^{- 1} \right)}{{F_{{4{({2^{k - 1} - j - 1})}},1}^{k}(z)}.}}} & (119)\end{matrix}$From (82) in which z is replaced with z⁻¹ and from the definition of m,we obtain that:

$\begin{matrix}{{{F_{{4m} + 1}^{k + 1}(z)} = {\left. {F_{{8{({2^{k - 1} - j - 1})}} + 1}^{k + 1}\left( z^{- 1} \right)}\Leftrightarrow{F_{{4m} + 1}^{k + 1}(z)} \right. = {F_{{4{({2^{k} - m - 1})}} + 1}^{k + 1}\left( z^{- 1} \right)}}},} & (120)\end{matrix}$for m=1,3,5, . . . , 2^(k−1)−1. Thus combining the results of equations(117) and (120), we have:

$\begin{matrix}{{{F_{{4m} + 1}^{k + 1}(z)} = {{{F_{{4{({2^{k} - m - 1})}} + 1}^{k + 1}\left( z^{- 1} \right)}\mspace{14mu}{for}\mspace{14mu} m} = 0}},1,2,\ldots\mspace{11mu},{2^{k - 1} - 1.}} & (121)\end{matrix}$For the prediction filters of the form

F_(4m + 2)^(k + 1),with m=0,1, . . . , 2^(k−1)−1, we follow the same reasoning as before,hence the symmetry properties are derived separately for m odd or even.For the case of m even, we denote m=2j, with 2j=0,2,4, . . . , 2^(k−1)−2equivalent to j=0,1,2, . . . , 2^(k−2)−1, and we use equations (83) and(95), obtaining:

$\begin{matrix}{{F_{{4m} + 2}^{k + 1}(z)} = {{F_{{8j} + 2}^{k + 1}(z)} = {\left. {z^{- 1}{F_{2}^{1}(z)}{F_{{4j},1}^{k}(z)}}\Leftrightarrow{F_{{4m} + 2}^{k + 1}(z)} \right. = {{{zF}_{2}^{1}\left( z^{- 1} \right)}{{F_{{4j},1}^{k}(z)}.}}}}} & (122)\end{matrix}$Since j=0,1, . . . , 2^(k−2)−1, we substitute (108) in equation (122)and we get:

F_(4m + 2)^(k + 1)(z) = z²F₂¹(z⁻¹)F_(4(2^(k − 1) − j − 1), 0)^(k)(z⁻¹)From (87) in which z is replaced with z⁻¹ and from the definition of m,we obtain that:

$\begin{matrix}{{{F_{{4m} + 2}^{k + 1}(z)} = {\left. {z^{2}{F_{{8{({2^{k - 1} - j - 1})}} + 6}^{k + 1}\left( z^{- 1} \right)}}\Leftrightarrow{F_{{4m} + 2}^{k + 1}(z)} \right. = {z^{2}{F_{{4{({2^{k} - m - 1})}} + 2}^{k + 1}\left( z^{- 1} \right)}}}},} & (123)\end{matrix}$for m=0,2,4, . . . , 2^(k−1)−2.For the case of m odd, we denote m=2j+1 with 2j+1=1,3,5, . . . ,2^(k−1)−1 equivalent to j=0,1,2, . . . , 2^(k−2)−1. Thus, for this casewe use equations (87) and (95), and we get:

$\begin{matrix}{{F_{{4m} + 2}^{k + 1}(z)} = {{F_{{8j} + 6}^{k + 1}(z)} = {\left. {{F_{2}^{1}(z)}{F_{{4j},0}^{k}(z)}}\Leftrightarrow{F_{{4m} + 2}^{k + 1}(z)} \right. = {z^{2}{F_{2}^{1}\left( z^{- 1} \right)}{{F_{{4j},0}^{k}(z)}.}}}}} & (124)\end{matrix}$Since j=0,1, . . . , 2^(k−2)−1, we substitute (107) in equation (124)and we obtain that:

F_(4m + 2)^(k + 1)(z) = z³F₂¹(z⁻¹)F_(4(2^(k − 1) − j − 1), 1)^(k)(z⁻¹).From (83) in which z is replaced with z⁻¹ and from the definition of m,we obtain:

$\begin{matrix}{{{F_{{4m} + 2}^{k + 1}(z)} = {\left. {z^{2}{F_{{8{({2^{k - 1} - j - 1})}} + 2}^{k + 1}(z)}}\Leftrightarrow{F_{{4m} + 2}^{k + 1}(z)} \right. = {z^{2}{F_{{4{({2^{k} - m - 1})}} + 2}^{k + 1}\left( z^{- 1} \right)}}}},} & (125)\end{matrix}$for m=1,3,5, . . . , 2^(k−1)−1. Thus combining the results of equations(123) and (125) we have:

$\begin{matrix}{{{F_{{4m} + 2}^{k + 1}(z)} = {{z^{2}{F_{{4{({2^{k} - m - 1})}} + 2}^{k + 1}\left( z^{- 1} \right)}\mspace{14mu}{for}\mspace{14mu} m} = 0}},1,2,\ldots\mspace{11mu},{2^{k - 1} - 1.}} & (126)\end{matrix}$Finally, for the filters of the form

F_(4m + 3)^(k + 1),with m=0,1, . . . , 2^(k−1)−1 we follow the same approach as previously.For the case of in even, we denote m=2j with j=0,1,2, . . . , 2^(k−2)−1.Hence, for this case we use equations (84) and (96):

$\begin{matrix}{{F_{{4m} + 3}^{k + 1}(z)} = {{F_{{8j} + 3}^{k + 1}(z)} = {\left. {{F_{{4j},0}^{k}(z)} + {z^{- 1}{F_{3}^{1}(z)}{F_{{4j},1}^{k}(z)}}}\Leftrightarrow{F_{{4m} + 3}^{k + 1}(z)} \right. = {{F_{{4j},0}^{k}(z)} + {{F_{3}^{1}\left( z^{- 1} \right)}{{F_{{4j},1}^{k}(z)}.}}}}}} & (127)\end{matrix}$Since j=0,1, . . . , 2^(k−2)−1, we substitute (107) and (108) inequation (127) and we get:

F_(4m + 3)^(k + 1)(z) = z(F_(4(2^(k − 1) − j − 1), 1)^(k)(z⁻¹) + F₃¹(z⁻¹)F_(4(2^(k − 1) − j − 1), 0)^(k)(z⁻¹))From (88) in which z is replaced with z⁻¹ and from the definition of m,we obtain:

$\begin{matrix}{{{F_{{4m} + 3}^{k + 1}(z)} = {\left. {{zF}_{{8{({2^{k - 1} - j - 1})}} + 7}^{k + 1}\left( z^{- 1} \right)}\Leftrightarrow{F_{{4m} + 3}^{k + 1}(z)} \right. = {z^{- 1}{F_{{4{({2^{k} - m - 1})}} + 3}^{k + 1}\left( z^{- 1} \right)}}}},{{{for}\mspace{14mu} m} = 0},2,4,\ldots\mspace{11mu},{2^{k - 1} - 2.}} & (128)\end{matrix}$For the case of m odd, we consider m=2j+1 with j=0,1,2, . . . ,2^(k−2)−1. Hence, by equations (88) and (96):

$\begin{matrix}{{F_{{4m} + 3}^{k + 1}(z)} = {{F_{{8j} + 7}^{k + 1}(z)} = {\left. {{F_{{4j},1}^{k}(z)} + {{F_{3}^{1}(z)}{F_{{4j},0}^{k}(z)}}}\Leftrightarrow{F_{{4m} + 3}^{k + 1}(z)} \right. = {{F_{{4j},1}^{k}(z)} + {{{zF}_{3}^{1}\left( z^{- 1} \right)}{{F_{{4j},0}^{k}(z)}.}}}}}} & (129)\end{matrix}$Since j=0,1, . . . , 2^(k−2)−1, we can substitute (107) and (108) inequation (129) and we get:

F_(4m + 3)^(k + 1)(z) = z(F_(4(2^(k − 1) − j − 1), 0)^(k)(z⁻¹) + zF₃¹(z⁻¹)F_(4(2^(k − 1) − j − 1), 1)^(k)(z⁻¹)).From (84) in which z is replaced with z⁻¹ and from the definition of m,we obtain:

$\begin{matrix}{{{F_{{4m} + 3}^{k + 1}(z)} = {\left. {{zF}_{{8{({2^{k - 1} - j - 1})}} + 3}^{k + 1}\left( z^{- 1} \right)}\Leftrightarrow{F_{{4m} + 3}^{k + 1}(z)} \right. = {{zF}_{{4{({2^{k} - m - 1})}} + 3}^{k + 1}\left( z^{- 1} \right)}}},} & (130)\end{matrix}$for m=1,3,5 . . . , 2^(k−1)−1. Thus, by combining the results ofequations (128) and (130) we have:

$\begin{matrix}{{{F_{{4m} + 3}^{k + 1}(z)} = {{{{zF}_{{4{({2^{k} - m - 1})}} + 3}^{k + 1}\left( z^{- 1} \right)}\mspace{14mu}{for}\mspace{14mu} m} = 0}},1,2,\ldots\mspace{11mu},{2^{k - 1} - 1.}} & (131)\end{matrix}$Equations (115), (121), (126) (131) can be derived from equations(89)-(92) by replacing k with k+1 Thus the symmetry propositionsP_(S)(k+1) are true. This end the proof of the induction, that is thesymmetry propositions P_(S)(E) are true for any level E, with E>1.

As a practical example of the symmetry properties, the filter taps ofthe prediction filters of the 9/7 filter-pair for decomposition levels1,2,3 are shown in Table II, III and IV. These properties, expressed bythe set of equations (89)-(92), allow the reduction of the necessarymultiplications for the complete-to-overcomplete transform derivation.This complexity analysis is given below.

As explained above, the level-by-level calculation of the overcompleterepresentation from the critically-sampled pyramid can be performed byusing two techniques: the LL-LBS method described above and theprediction-filters method in accordance with the invention. Eachtechnique operates in two modes depending on the current decompositionlevel. The first mode is the full-overcomplete (FO) transform-productionmode, where the current decomposition level is the coarsest-resolutionrepresentation, therefore the low and the high-frequency subbands areproduced (level 3 in the particular example of FIG. 3). The second isthe high-frequency overcomplete (HFO) transform-production mode, wherethe current decomposition level is an intermediate-resolution level, andas a consequence, only the high-frequency subbands of this resolutionneed to be computed. In order to estimate the complexity of eachtechnique, two factors are taken into account. The first is thenecessary number of multiplication operations, which corresponds to thecomputational complexity of each method, since multiplication is thedominant operation in convolutional systems. The second is the delay forthe production of the results of every level, which can be estimatedunder some assumptions concerning the degree of parallelism achievablein the implementation of every technique. Since systems with a highdegree of parallelism are assumed and designs with the minimum amount ofmultiplications are chosen for both methods, the results of this sectionare more realistic for custom-hardware rather than processor-basedsolutions; in the latter, a lower level of parallelism is feasible andthe minimization of MAC operations is the critical issue in thecomplexity reduction.

The following section focuses on the complexity analysis of the 1-Dapplication of both methods. Moreover, following the extension of theprediction-filters to 2-D decompositions, it can be shown that thelevel-by-level application of the 2-D prediction filters is completelyseparable to the row and column filtering with the corresponding 1-Dkernels. This matter is further discussed below.

Required Number of Multiplication Operations

We assume the general case of a biorthogonal filter-pair with T_(H),T_(G) denoting the number of taps for the filters H, {tilde over (G)},and G, {tilde over (H)} respectively. In addition we restrict ourfurther presention to the case of point-symmetric filters, though theinvention is not limited thereto; this option is mainly motivated by thefact that this sub-family of biorthogonal filters typically gives verygood results in still-image and video coding.

For both methods we assume convolutional implementations. A liftingimplementation is possible for the prediction filters, if thefactorization algorithm of Sweldens is applied. Assuming convolutionalimplementation, for every application of the filter kernels we need toperform

$\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor\mspace{14mu}{and}\mspace{14mu}\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor$multiplications because of the point-symmetry property of filters H,G.Notice that, both for the LL-LBS and the prediction-filters method, theclassical point-symmetric extension (mirroring) is assumed for thesignal edges. For an one-level wavelet decomposition or reconstructionof an N-point signal, the required number of multiplications is:

$\begin{matrix}{{{X_{C}(N)} = {{\Psi_{C}(N)} = {\left( {\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor\; + \left\lfloor \frac{T_{G} + 1}{2} \right\rfloor} \right)\frac{N}{2}}}},} & (132)\end{matrix}$where X_(C)(N),Ψ_(C)(N) denote the number of multiplications needed toperform a complete decomposition and reconstruction of N samplesrespectively. For a decomposition where only the low-frequency (average)subband is produced, the required number of multiplications for anN-sample signal is:

$\begin{matrix}{{X_{A}(N)} = {\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor\;{\frac{N}{2}.}}} & (133)\end{matrix}$Similarly, for a decomposition where only the high-frequency (detail)subband is produced, the number of multiplications is:

$\begin{matrix}{{X_{D}(N)} = {\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor\;{\frac{N}{2}.}}} & (134)\end{matrix}$Finally, for a reconstruction from the low-frequency (average) subbandonly (only the number of non-zero multiplication operations is takeninto account), the required multiplications for an N-sample outputsignal are:

$\begin{matrix}{{\Psi_{A}(N)} = {\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor{\frac{N}{2}.}}} & (135)\end{matrix}$By utilizing equations (132)-(135), we can calculate the required numberof multiplications needed to derive the subbands A_(i) ^(k),D_(i) ^(k)starting from subbands A₀ ^(k) and D₀ ^(k), with k and iε[1,2^(k)−1]denoting the decomposition level and the subband index respectively. Asexplained before, in order to ensure spatial scalability, allhigher-resolution levels are assumed non-present during this procedure,hence D₀ ^(l)=0 for any lε[1,k−1].

As described above k inverse transforms should be performed toreconstruct the low-frequency subband (or the input signal) in the caseof the LL-LBS method; for an example of k=3. Then, by performing anumber of forward transforms retaining the even or odd polyphasecomponents, all the subbands of the overcomplete representation at levelk are produced. In general, starting from the subbands of thecritically-sampled decomposition of an N-point sequence at level k, thenumber of multiplications required to perform the inverse transforms is:

$\begin{matrix}{{{\Psi_{C}\left( \frac{N}{2^{k - 1}} \right)} + {\Psi_{A}\left( \frac{N}{2^{k - 2}} \right)} + {\Psi_{A}\left( \frac{N}{2^{k - 3}} \right)} + \ldots + {\Psi_{A}(N)}} = {{N\left\lbrack {{\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor 2^{- k}} + {\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor\left( {1 - 2^{- k}} \right)}} \right\rbrack}.}} & (136)\end{matrix}$The multiplications required to perform the forward transforms, whenoperating in the FO-mode, are:

$\begin{matrix}{\begin{matrix}{{{\left( {2 - 1} \right){X_{A}(N)}} + {\left( {4 - 1} \right){X_{A}\left( \frac{N}{2} \right)}} + \ldots + {\left( {2^{k - 1} - 1} \right){X_{A}\left( \frac{N}{2^{k - 2}} \right)}} + {\left( {2^{k} - 1} \right){X_{C}\left( \frac{N}{2^{k - 1}} \right)}}} =} \\{= {N\left\lbrack {{\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor \cdot \left( {k - 1 + 2^{- k}} \right)} + {\left( {1 - 2^{- k}} \right)\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor}} \right\rbrack}}\end{matrix}.} & (137)\end{matrix}$The total number of multiplications performed with the LL-LBS method inorder to calculate all the subbands of level k in the FO-mode can bederived by the sum of equations (136) and (137), given by:

$\begin{matrix}{{M_{{{LL}\text{-}{LBS}},{FO}}(k)} = {{N\left\lbrack {{\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor \cdot \left( {k - 1 + 2^{1 - k}} \right)} + {\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor \cdot \left( {2 - 2^{1 - k}} \right)}} \right\rbrack}.}} & (138)\end{matrix}$When operating in the HFO-mode, the number of multiplications for theforward transforms of the current decomposition level k is modified,since for this level only the high-frequency subbands are produced:

$\begin{matrix}{\begin{matrix}{{{\left( {2 - 1} \right){X_{A}(N)}} + {\left( {4 - 1} \right){X_{A}\left( \frac{N}{2} \right)}} + \ldots + {\left( {2^{k - 1} - 1} \right){X_{A}\left( \frac{N}{2^{k - 2}} \right)}} + {\left( {2^{k} - 1} \right){X_{D}\left( \frac{N}{2^{k - 1}} \right)}}} =} \\{= {N\left\lbrack {{\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor \cdot \left( {k - 2 + 2^{1 - k}} \right)} + {\left( {1 - 2^{- k}} \right)\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor}} \right\rbrack}}\end{matrix}.} & (139)\end{matrix}$Hence, the total multiplication budget is in this case:

$\begin{matrix}{{M_{{{LL}\text{-}{LBS}},{HFO}}(k)} = {{N\left\lbrack {{\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor \cdot \left( {k - 2 + {3 \cdot 2^{- k}}} \right)} + {\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor\left( {2 - 2^{1 - k}} \right)}} \right\rbrack}.}} & (140)\end{matrix}$In order to derive for the prediction-filters method the necessarynumber of multiplications for the level-by-level calculation of theovercomplete representation in the FO and HFO modes, we need to specifythe necessary convolutions. For the level-by-level construction of levelk with k≧2, we only have the subbands A₀ ^(k),D₀ ^(k), since D₀ ^(l)=0for any lε[1,k−1]. As a consequence, the sets of equations (6) and (8)are simplified to the following expressions:A ₁ ^(k) [n]=F ₀ ¹ [n]*A ₀ ^(k) [n]+F ₁ ¹ [n]*D ₀ ^(k) [n]D ₁ ^(k) [n]=F ₂ ¹ [n]*A ₀ ^(k) [n]+F ₃ ¹ [n]*D ₀ ^(k) [n],  (141)with k≧2, and

$\begin{matrix}{\begin{matrix}{{A_{{2i_{L}} + 2^{l}}^{k}\lbrack n\rbrack} = {{{F_{8i_{L}}^{l + 1}\lbrack n\rbrack}*{A_{0}^{k}\lbrack n\rbrack}} + {{F_{{8i_{L}} + 1}^{l + 1}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}}}} \\{{D_{{2i_{L}} + 2^{l}}^{k}\lbrack n\rbrack} = {{{F_{{8i_{L}} + 2}^{l + 1}\lbrack n\rbrack}*{A_{0}^{k}\lbrack n\rbrack}} + {{F_{{8i_{L}} + 3}^{l + 1}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}}}} \\{{A_{{2i_{L}} + 2^{l} + 1}^{k}\lbrack n\rbrack} = {{{F_{{8i_{L}} + 4}^{l + 1}\lbrack n\rbrack}*{A_{0}^{k}\lbrack n\rbrack}} + {{F_{{8i_{L}} + 5}^{l + 1}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}}}} \\{{D_{{2i_{L}} + 2^{l} + 1}^{k}\lbrack n\rbrack} = {{{F_{{8i_{L}} + 6}^{l + 1}\lbrack n\rbrack}*{A_{0}^{k}\lbrack n\rbrack}} + {{F_{{8i_{L}} + 7}^{l + 1}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}}}}\end{matrix},} & (142)\end{matrix}$with i_(L)ε[0,2^(l−1)−1], for every lε[1,k−2] and k≧3. Also,

$\begin{matrix}{\begin{matrix}{{A_{{2i_{R}} + 2^{k - 1}}^{k}\lbrack n\rbrack} = {{{F_{8i_{R}}^{k}\lbrack n\rbrack}*{A_{0}^{k}\lbrack n\rbrack}} + {{F_{{8i_{R}} + 1}^{k}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}}}} \\{{D_{{2i_{R}} + 2^{k - 1}}^{k}\lbrack n\rbrack} = {{{F_{{8i_{R}} + 2}^{k}\lbrack n\rbrack}*{A_{0}^{k}\lbrack n\rbrack}} + {{F_{{8i_{R}} + 3}^{k}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}}}} \\{{A_{{2i_{R}} + 2^{k - 1} + 1}^{k}\lbrack n\rbrack} = {{{F_{{8i_{R}} + 4}^{k}\lbrack n\rbrack}*{A_{0}^{k}\lbrack n\rbrack}} + {{F_{{8i_{R}} + 5}^{k}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}}}} \\{{D_{{2i_{R}} + 2^{k - 1} + 1}^{k}\lbrack n\rbrack} = {{{F_{{8i_{R}} + 6}^{k}\lbrack n\rbrack}*{A_{0}^{k}\lbrack n\rbrack}} + {{F_{{8i_{R}} + 7}^{k}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}}}}\end{matrix},} & (143)\end{matrix}$with i_(R)ε[0,2^(k−2)−1],i_(R)εZ and k≧2.

Equations (141), (142) and (143) represent the calculation of the“left-half” and “right-half” of the overcomplete pyramid as seen fromFIG. 5. Notice that each of the two subbands A₀ ^(k),D₀ ^(k) contains

$\frac{N}{2^{k}}$coefficients. From (141), (142), (143), it can be noticed that in thelevel-by-level framework, the calculation of the overcomplete subbandswith the prediction filters described by the sets of equations (6) and(8) are reduced to a compact form that depends only on thecritically-sampled subbands of current level, and on the set of thefilters F_(m) ^(l) with lε[1k],lεZ and mε[0,2^(l+1)−1],mεZ. As anexample, by replacing k=3 one can extract from (141), (142) and (143)the necessary convolutions for the particular example of thelevel-by-level derivation of the subbands of level 3 via theprediction-filters method. This particular example is illustrated inFIG. 4.

The total calculation budget can be reduced if instead of performingdirectly the convolutions in (142) and (143) with the filters

F_(8i + {0,  , 7})^(l + 1),lε[1,k−1], i=0,1, . . . , 2^(l−2)−1, one replaces their expressions withtheir equivalent form given in equations (9)-(16). From these equations,it can be seen that the direct convolution with the filters

F_(8i + (0,  , 7))^(l + 1)can be replaced by convolutions with the filters

F_(4i, 0)^(l − 1), F_(4i, 1)^(l − 1),coupled with the reuse of the results of convolution with the filters F₀¹, F₁ ¹, F₂ ¹, F₃ ¹.This means that an update structure can be implemented so that theinitial convolutions performed with the filters F₀ ¹, F₁ ¹, F₂ ¹, F₃ ¹in the calculation of the subbands A₁ ^(k),D₁ ^(k) given by (141) arereused in the calculations performed in (142) and (143). Thisobservation is formalized in Table V, where for each of the equations(9)-(16), the necessary convolutions and the corresponding number ofmultiplications are given as a function of the number of input samples(N), the decomposition level (k) and the number of taps of the usedfilter, denoted by T

T_(F_(4i, q)^(l − 1))with q=0,1. Notice from Table V that the reused convolutions ofequations (9)-(16) are: [F₀ ¹[n]*A₀ ^(k)[n]], [F₁ ¹[n]*D₀ ^(k)[n]], [F₂¹[n]*A₀ ^(k)[n]] and [F₃ ¹[n]*D₀ ^(k)[n]]. Notice also that [F₀ ¹[n]*A₀^(k)[n]] is equivalent to −[F₃ ¹[n]*A₀ ^(k)[n]], as it comes out fromequations (17), (18), and hence can be used also for the convolution F₃¹[n]*A₀ ^(k)[n] which appears in (9).

By summing up the multiplications reported in Table V and the necessarymultiplications required for the filters of the first level, one canderive the number of the multiplication operations needed to calculatein the FO mode the subbands for every level k for an N-point inputsignal:

$\begin{matrix}{M_{{P - {Filters}},{FO}} = {{\frac{N}{2^{k}}\left( {{\sum\limits_{l = 2}^{k}\;{\sum\limits_{i = 0}^{2^{i - 2} - 1}\;{6\left( {T_{F_{{4i},0}^{i - 1}} + T_{F_{{4i},1}^{i - 1}}} \right)}}} + {\sum\limits_{i = 0}^{3}\; T_{F_{i}^{1}}}} \right)} = {\frac{N}{2^{k}}{\left( {{6{\sum\limits_{l = 2}^{k}{\sum\limits_{i = 0}^{2^{i - 2} - 1}T_{F_{4i}^{i - 1}}}}} + {\sum\limits_{i = 0}^{3}T_{F_{i}^{1}}}} \right) \cdot}}}} & (144)\end{matrix}$In the HFO mode, where only the high-frequency subbands of theovercomplete transform are calculated, only the filters F_(8i) _(L) ₊₂^(l+1), F_(8i) _(L) ₊₃ ^(l+1), F_(8i) _(L) ₊₆ ^(l+1), F_(8i) _(L) ₊₇^(l+1) and F_(8i) _(R) ₊₂ ^(k),F_(8i) _(R) ₊₃ ^(k), F_(8i) _(R) ₊₆^(k),F_(8i) _(R) ₊₇ ^(k) ∀i_(L),i_(R) have to be used for the“left-half” and “right-half” parts, as seen from (142) and (143). Forevery value of k, this corresponds to filters F_(8i+2) ^(k), F_(8i+3)^(k) and F_(8i+6) ^(k), F_(8i+7) ^(k) of equations (11), (12) and(15),(16). Thus the situation is limited to the second and the fourthrow of Table V and to filters F₂ ¹ and F₃ ¹ for the first decompositionlevel (equation (141)). As a result, if one assumes that the system isoperating in the HFO mode, equation (144) is modified to:

$\begin{matrix}{M_{{P - {Filters}},{HFO}} = {{\frac{N}{2^{k}}\left( {{\sum\limits_{l = 2}^{k}\;{\sum\limits_{i = 0}^{2^{i - 2} - 1}{3\left( {T_{F_{{4i},0}^{i - 1}} + T_{F_{{4i},1}^{i - 1}}} \right)}}} + T_{F_{2}^{1}} + T_{F_{3}^{1}}} \right)} = {\frac{N}{2^{k}}{\left( {{\underset{l = 2}{\overset{k}{3\sum}}\;{\sum\limits_{i = 0}^{2^{i - 2} - 1}T_{F_{4i}^{i - 1}}}} + T_{F_{2}^{1}} + T_{F_{3}^{1}}} \right).}}}} & (145)\end{matrix}$For the prediction filters of level 1, one can reduce the number ofmultiplications by exploiting the point-symmetry (or half-pointsymmetry) properties of these filters, expressed in equations (93)-(98).In addition, the general symmetry-properties for the prediction filtersof point-symmetric biorthogonal filter-pairs can be used to reduce thenumber of multiplications for the higher decomposition levels as well,as explained in the following.

Convoluting an input sequence I[n] with two FIR filters U[n] and W[n]that are linked with the relationship W(z)=z^(−a)U(z⁻¹) requires thesame multiplications but performed in a different order. This is acorollary from the fact that by representing the two filters in thefrequency domain, one has W(e^(jΩ))=e^(−jΩa)U(e^(jΩ))*, and hence thesequences U(e^(jΩ))·I(e^(jΩ)), U(e^(jΩ))*·I(e^(jΩ)) represent the samemultiplications but with alternating signs for the imaginary part (dueto the complex conjugate in the frequency domain). As a result, a memorystructure can be utilized to delay the intermediate results of theconvolution in the time domain for the parallel calculation of U[n]*I[n]and W[n]*I[n]. For an L-tap filter U[n], such a memory structure isshown in FIG. 6. Similar forms are commonly utilized for the efficientrealization of FIR orthogonal filter-banks. Using such memory structuresfor the parallel application of the prediction filters, the requirednumber of multiplications can be reduced by half, for every set ofprediction filters F₁ ^(l). This can be seen from the example of FIG. 4,where, if four memory structures are utilized for the application offilters F₀ ²,F₁ ²,F₂ ²,F₃ ² to the input sequences A₀ ³,D₀ ³, theresults of the application of filters F₄ ²,F₅ ²,F₆ ²,F₇ ² are obtainedas well, due to the symmetry properties; see equations (99)-(102).Hence, subbands A₂ ³,D₂ ³ and A₃ ³,D₃ ³ are created in parallel byapplying only one quadruplet of filters implemented with four memorystructures similar to the one seen in FIG. 6. In the same fashion, theapplication of filters F₀ ³,F₁ ³,F₂ ³,F₃ ³,F₄ ³,F₅ ³,F₆ ³,F₇ ³ in FIG. 4creates the results of F₁₂ ³,F₁₃ ³,F₁₄ ³,F₁₅ ³,F₈ ³,F₉ ³,F₁₀ ³,F₁₁ ³respectively; see equations (89)-(92) with k=3.

Generalizing this property, for every level k, the convolutions offilters F₀ ¹, F₁ ¹, F₂ ¹, F₃ ¹ and filters F_(j) ^(l) or F₂ _(l) _(+j)^(l) with lε[2,k] and ∀l: jε[0,2^(l)−1] with the input subbands A₀^(l),D₀ ^(l) suffice for the calculation of all the subbands of theovercomplete representation of this level. Equivalently, a mixedcalculation-order can be selected, i.e. the convolutions with filtersF_(4m) ^(l), F_(4m+1) ^(l), F_(4m+2) ^(l), F_(4m+3) ^(l) can be used toproduce filters F₄₍₂ _(l−1) _(−m−1)) ^(l), F₄₍₂ _(l−1) _(−m−1)+1) ^(l),F₄₍₂ _(l−1) _(−m−1)+2) ^(l), F₄₍₂ _(l−1) _(−m−1)+3) ^(l) respectivelyfor the even values of m (and zero) and vice-versa for the odd values ofm, with m=0,1, . . . , 2^(l−2)−1 and lε[2,k]. For example, for thecalculation of the overcomplete transform of level 3, the convolutionswith filters F₀ ³,F₁ ³,F₂ ³,F₃ ³ and F₈ ³, F₉ ³, F₁₀ ³, F₁₁ ³ canprovide the results of filtering with F₁₂ ³,F₁₃ ³,F₁₄ ³, F₁₅ ³ and F₄³,F₅ ³,F₆ ³,F₇ ³ respectively.

However, as mentioned before, in order to reuse the previously producedresults, the implementation of the convolutions with filters F_(j) ^(l)and F₂ _(l) _(+j) ^(l) is based on the update structure shown in Table Vand not on the direct application of the filters. Thus, filtering withF_(j) ^(l) and F₂ _(l) _(+j) ^(l) is performed by the separateapplications of filters F_(4i,0) ^(l−1)(z), F_(4i,1) ^(l−1)(z), as seenin Table V. As a result, in order for the multiplication-reductions tobe valid in this update-structure implementation, it should be shownthat all convolutions with the filters seen in Table V can produce theconvolutions with the time-inverse filters as well, without additionalmultiplications. This is shown in the next paragraph.

By replacing z with z⁻¹ in equations (9)-(16), the time-inversedupdate-structure of the prediction filters can be written:

$\begin{matrix}{{{F_{8i}^{k}\left( z^{- 1} \right)} = {{F_{{4i},0}^{k - 1}\left( z^{- 1} \right)} - {z\;{F_{3}^{1}\left( z^{- 1} \right)}{F_{{4i},1}^{k - 1}\left( z^{- 1} \right)}}}},} & (146) \\{{{F_{{8i} + 1}^{k}\left( z^{- 1} \right)} = {z\;{F_{1}^{1}\left( z^{- 1} \right)}{F_{{4i},1}^{k - 1}\left( z^{- 1} \right)}}},} & (147) \\{{{F_{{8i} + 2}^{k}\left( z^{- 1} \right)} = {z\;{F_{2}^{1}\left( z^{- 1} \right)}{F_{{4i},1}^{k - 1}\left( z^{- 1} \right)}}},} & (148) \\{{F_{{8i} + 3}^{k}\left( z^{- 1} \right)} = {{F_{{4i},0}^{k - 1}\left( z^{- 1} \right)} + {z\;{F_{3}^{1}\left( z^{- 1} \right)}{F_{{4i},1}^{k - 1}\left( z^{- 1} \right)}}}} & (149) \\{{{F_{{8i} + 4}^{k}\left( z^{- 1} \right)} = {{F_{{4i},1}^{k - 1}\left( z^{- 1} \right)} + {{F_{0}^{1}\left( z^{- 1} \right)}{F_{{4i},0}^{k - 1}\left( z^{- 1} \right)}}}},} & (150) \\{{{F_{{8i} + 5}^{k}\left( z^{- 1} \right)} = {{F_{1}^{1}\left( z^{- 1} \right)}{F_{{4i},0}^{k - 1}\left( z^{- 1} \right)}}},} & (151) \\{{{F_{{8i} + 6}^{k}\left( z^{- 1} \right)} = {{F_{2}^{1}\left( z^{- 1} \right)}{F_{{4i},0}^{k - 1}\left( z^{- 1} \right)}}},} & (152) \\{{F_{{8i} + 7}^{k}\left( z^{- 1} \right)} = {{F_{{4i},1}^{k - 1}\left( z^{- 1} \right)} + {{F_{3}^{1}\left( z^{- 1} \right)}{F_{{4i},0}^{k - 1}\left( z^{- 1} \right)}}}} & (153)\end{matrix}$For the filters seen in the right parts of expressions (146)-(153), theapplication of z^(−a)F_(4i,0) ^(l−1)(z⁻¹)·I₁, z^(−a) F_(4i,1)^(l−1)(z⁻¹)·I₂, with a=0,−1, to any inputs I₁,I₂ can be performed byF_(4i,0) ^(l−1)(z)·I₁, F_(4i,1) ^(l−1)(z)·I₂ with the use of two memorystructures such as the one seen in FIG. 6. In addition, based on thesymmetry properties shown in equations (93)-(97), any convolution withfilters F₀ ¹(z⁻¹),F₁ ¹(z⁻¹),F₂ ¹(z⁻¹),F₃ ¹(z⁻¹) is equivalent to theconvolution with F₀ ¹, F₁ ¹,F₂ ¹,F₃ ¹ (with the appropriate delays). Asa result, all the filter-applications shown in the time-inversedupdate-structure of (146)-(153) can be implemented by the convolutionsshown in Table V and a number of memory structures such as the one shownin FIG. 6; hence the update-structure implementation can provide theconvolutions with the time-inversed filters with no additionalarithmetic operations. Summarizing, since the update-structureimplementation of filters F_(j) ^(l)(z) can provide the results offilters F_(j) ^(l)(z⁻¹), based on the prediction-filter symmetries, theresults of filters F_(j) ^(l)(z⁻¹) provide the convolutions with F₂ _(l)_(+j) ^(l)(z).

Exploiting this symmetry by selecting the filters to implement accordingto the mixed calculation-order that was mentioned before, the number ofmultiplications in the FO-mode with the prediction-filters method isreduced to the half of Table V. By choosing to implement equations(9)-(12), the required multiplications are:

$\begin{matrix}{M_{{P - {Filters}},{sym},{FO}} = {\frac{N}{2^{k}}{\left( {{\underset{l = 2}{\overset{k}{2\sum}}\;{\sum\limits_{i = 0}^{2^{i - 2} - 1}\left( {T_{F_{{4i},0}^{i - 1}} + {2T_{F_{{4i},1}^{i - 1}}}} \right)}} + {\sum\limits_{i = 0}^{3}\;\left\lfloor \frac{T_{F_{i}^{1}} + 1}{2} \right\rfloor}} \right).}}} & (154)\end{matrix}$Similarly, for the HFO-mode, the implementation of equations (11), (12)requires:

$\begin{matrix}{M_{{P - {Filters}},{sym},{HFO}} = {\frac{N}{2^{k}}{\left( {{\sum\limits_{l = 2}^{k}\;{\sum\limits_{i = 0}^{2^{i - 2} - 1}\left( {T_{F_{{4i},0}^{i - 1}} + {2T_{F_{{4i},1}^{i - 1}}}} \right)}} + \;\left\lfloor \frac{T_{F_{2}^{1}} + 1}{2} \right\rfloor + \;\left\lfloor \frac{T_{F_{3}^{1}} + 1}{2} \right\rfloor} \right).}}} & (155)\end{matrix}$Table VI shows for both the FO and the HFO-modes, the number of taps forthe filters

F_(4i)^(l − 1), F_(4i, 0)^(l − 1), F_(4i, 1)^(l − 1)for the typical 9/7 biorthogonal filter-pair, with lε[2,4]. This tablereports also the required number of multiplications for the LL-LBSmethod and for the prediction-filters method for the construction of allsubbands of level l as calculated by equations (138), (140) and (154),(155) respectively. The last column of Table VI shows the percentage ofthe reduction in the multiplication budget.

The numbers of multiplications shown in parenthesis correspond to anapproximation of the prediction filters where, for the filters of theupdate structure of every level, all taps smaller than a threshold areset to zero. In this way, the size of the filters of each level isreduced, while a good approximation of the final result is obtained.This technique cannot be applied in the LL-LBS approach since the tapsof the biorthogonal filter-pairs do not have magnitudes below the chosenthresholds. Table VIII shows the values used for the thresholds and theresulting maximum mean-square error (MMSE) between the results obtainedwith the original and thresholded prediction filters when applied in the2-D (row-column) manner to the 8-bit images of the JPEG-2000 test-set.It can be observed from the MMSE values that the chosen method forthresholding has a minimal effect on the results of the predictionfilters while it reduces significantly the computational load.

Delay for the Calculation of the Subbands of Decomposition Level k.

The delay occurring in the calculation of the subbands of theovercomplete pyramid of decomposition level k using theprediction-filters and the LL-LBS methods is now presented. As in theprevious sections, we discuss the case of biorthogonal, point-symmetricfilter-pairs. Consider that the two methods are implemented in a systemwhere one application of a filter-kernel on an input sequence requiresa_(LBS) processing cycles for the LL-LBS method and a_(PF) processingcycles for the prediction-filters method. Furthermore, to diminish theside effects of scheduling algorithms for the multiple filteringoperations, we assume the case of high parallelism, where onefilter-kernel per required convolution is present. In this way, everyfiltering application initiates as soon as sufficient input is present.Moreover, to facilitate the description, the delay resulting from thestorage or retrieval of intermediate results is not taken into account.In addition, in this case of high-parallelism, the delay for thecalculations in the FO mode is equivalent to the one for the HFO mode,with the latter requiring less filter-kernels. Hence no distinction ismade between the two modes in the following, and the numbers for thefilter-kernels refer to the FO mode, which represents the worst casebetween the two.

Starting from the subbands A₀ ^(k),D₀ ^(k), the LL-LBS method performs kinverse transforms and

$\sum\limits_{l = 1}^{k}\;\left( {2^{l} - 1} \right)$forward transforms to produce the subbands of the overcompleterepresentation. See FIG. 3 for an example with k=3 and FIG. 5 whichrepresents the overcomplete pyramid in the general case of k+1decomposition levels. The cascade initiation of each of the inversetransforms at levels k−1,k−2, . . . , 1 requires

$\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor$input samples to be present at every level, so that mirroring and thefirst filter-kernel application can take place. Such an example is givenin the top-left part of FIG. 7 for the calculation of level k−1 fromlevel k. Notice from this figure that since the subbands D₀ ^(l) withlε[1,k−1] are not available, the samples of the low-frequency subbandsare interpolated with zeros at every level l.

After the initiation phase, all filter-kernels of every level work inparallel to perform the inverse transforms; this is equivalent to theinverse RPA algorithm of Vishwanath, but without considering thehigh-frequency subbands. In order to produce

$\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor$samples at every level, the filter-kernel of the previous level isapplied

$\left\lfloor \frac{\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor + 1}{2} \right\rfloor$times, as seen in the top-left of FIG. 7 with T_(G)=7. Similarly, forthe initiation of the entire forward transforms,

$\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor$input samples is present in the signal X and at the levels 1,2, . . . ,k−1. This implies

$\left\lfloor \frac{\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor + 1}{2} \right\rfloor$applications of the filter-kernel in the previous level, to initiate theforward transform in the current level. An example of the mirroring andinitiation of the forward transform of level 1 which is calculated fromthe signal x is given in the lower-left part of FIG. 7 with T_(H)=9.After this point, all filter-kernels of the LL-LBS system work inparallel. Hence, the latency occurring in the initiation phase of theproduction of all subbands of level k is:

$\begin{matrix}{{L_{{init},{{LL}\text{-}{LBS}}}(k)} = {\left\lbrack {{\left( {k - 1} \right)\left\lfloor \frac{\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor + 1}{2} \right\rfloor} + {k\left\lfloor \frac{\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor + 1}{2} \right\rfloor}} \right\rbrack \cdot {a_{LBS}.}}} & (156)\end{matrix}$The total time required for the completion of all subbands is determinedby the filtering of the sequence with the maximum length, since duringthis process the highest number of consecutive filter-kernelapplications occurs. The sequence with the maximum length in the pyramidof the overcomplete transform is the sequence of the reconstructedsignal X. The filtering of x requires

$\frac{N}{2}$applications of filter H to produce the subband A₁ ¹, as it can benoticed from the pictorial explanation given in the lower part of FIG.7.

To summarize, the latency occurring in the production of the firstcoefficient of this sequence is

$\left( {k - 1} \right)\left\lfloor \frac{\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor + 1}{2} \right\rfloor\; a_{LBS}$processing cycles (see equation (156)). The filtering of X requires

$\frac{N}{2}a_{LBS}$processing cycles to produce the subband A₁ ¹. After the completion ofthis operation, finalized with the mirroring of the subband border, thecoefficient

$a_{\frac{N}{4} - 1}^{1}$is produced. The remaining filtering applications for the followingdecomposition levels (levels 1, 2, . . . , k) require

$k\left\lfloor \frac{\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor + 1}{2} \right\rfloor a_{LBS}$processing cycles, so that, in the same manner, all the coefficients

$a_{\;^{\frac{N}{2^{l + 1}} - 1}}^{l}$with lε[2,k] are produced. As a result, the total delay of the LL-LBSsystem for the production of all the subbands of decomposition level kfor an N-point input sequence is:

$\begin{matrix}{{L_{{LL} - {LBS}}\left( {k,N} \right)} = {\quad{\quad{{\left\lbrack {{\left( {k - 1} \right)\left\lfloor \frac{\left\lfloor \frac{T_{G} + 1}{2} \right\rfloor + 1}{2} \right\rfloor} + \frac{N}{2} + {k\left\lfloor \frac{\left\lfloor \frac{T_{H} + 1}{2} \right\rfloor + 1}{2} \right\rfloor}} \right\rbrack a_{LBS}} = {\left( {{L_{{init},{{LL} - {LBS}}}(k)} + \frac{N}{2}} \right)a_{LBS}}}}}} & (157)\end{matrix}$For the prediction-filters method, the convolutions with the filters F₀²,F₁ ²,F₂ ²,F₃ ²,can be initiated in parallel for the calculation of thesubbands A₁ ^(k), D₁ ^(k). Subsequently, the filters that produce therest of the subbands of level k can also be applied in parallel usingthe update-structure implementation. The application of all thesefilter-kernels will occur as soon as enough coefficients from thesequences [F₀ ¹[n]*A₀ ^(k)[n]], [F₁ ¹[n]*D₀ ^(k)[n]], [F₂ ¹[n]*A₀^(k)[n]], [F₃ ¹[n]*D₀ ^(k)[n]] are calculated, so that the necessarymirroring in the update structure can be performed. In addition, a delayequal to the maximum filter-length is required so that the calculationsof the subbands that are produced by the time-inversed impulse responsesare initiated as well by the memory structures. Hence, the latency forthe initiation of all the convolutions is:

$\begin{matrix}{{{L_{{init},{P - {Filters}}}(k)} = {a_{PF} \cdot \left( {\left\lfloor \frac{{\max\left\{ {T_{F_{i,0}^{l}},T_{F_{i,1}^{l}},T_{F_{j}^{l}}} \right\}} + 1}{2} \right\rfloor + {\max\left\{ {T_{F_{i,0}^{l}},T_{F_{i,1}^{l}},T_{F_{j}^{l}}} \right\}}} \right)}},} & (158)\end{matrix}$with lε[2,k−1],iε[0,2^(l−2)−1],jε[0,3].Then, the delay for the completion of the process is simply:

$\begin{matrix}{{L_{P - {Filters}}\left( {k,N} \right)} = {{{L_{{init},{P - {Filters}}}(k)} + {\frac{N}{2^{k}}a_{PF}}} = {\left( {\left\lfloor \frac{{\max\left\{ {T_{F_{i,0}^{l}},T_{F_{i1}^{l}},T_{F_{j}^{l}}} \right\}} + 1}{2} \right\rfloor + {\max\left\{ {T_{F_{i,0}^{l}},T_{F_{i1}^{l}},T_{F_{j}^{l}}} \right\}} + \frac{N}{2^{k}}} \right){a_{PF}.}}}} & (159)\end{matrix}$The results of (157), (159) show that, under the same assumptions ofsystem parallelism, the complete to overcomplete transform derivation ofresolution level k for an N-point input sequence via theprediction-filters method achieves a delay proportional to

$\frac{N}{2^{k}}a_{PF}$processing cycles, while the LL-LBS approach produces the subbands ofthe same level with a fixed delay proportional to

$\frac{N}{2}{a_{LBS}.}$Table VII shows a comparison for a typical case whereN=512,T_(H)=9,T_(G)=7 for various decomposition levels. We notice fromthese results that a significant gain is achieved in the reduction ofthe system delay with the prediction-filters method.Extension to 2-D

The 2-D extension of the LL-LBS and the prediction-filters method can beintuitively performed by extending the 1-D application to a separateapplication to the rows of the input subbands and to the columns of theproduced results. Such extensions are due to the separability propertyof the DWT. For the level-by-level construction of the overcomplete DWTof an N×M input frame, the separable row and column application leads tothe same computational gains for the prediction-filters method incomparison to the LL-LBS approach as in the 1-D case. The delay of everymethod when applied in two dimensions is dependent on the level ofparallelism for each system. Since the prediction filters methodrequires only the application of a set of filters in the row and columndirections, under any practically-applicable level of parallelism, thismethod is expected to achieve a much lower delay for the production ofthe 2-D overcomplete transform than the 2-D LL-LBS method, whichconsists of a cascaded multirate filter-bank implementation. Similarly,every method can be implemented by two separate systems: one for the rowand one for the column processing. If the results after the productionof one row of every subband are immediately used for the continuation ofthe vertical filtering for all the columns of this subband, and thecolumn filtering completes when the next row is received, the delay forthe production of the results is simply the delay for the completion ofthe row-by-row filtering plus the delay for the completion of thecolumns' processing with the results of the last row. The columnprocessing begins after an initiation latency so that enoughcoefficients exist columnwise for the mirroring and for the initiationof the filter-applications required for every method. Since the processis separable, the initiation latency is equivalent to the one of the 1-Dcase. Hence, for the LL-LBS method, the required processing-cycles are:

$\begin{matrix}\begin{matrix}{{L_{{2D},{{LL} - {LBS}}}\left( {k,{N \times M}} \right)} = {\left( {N + \frac{L_{{init},{{LL} - {LBS}}}(k)}{a_{LBS}}} \right){L_{{LL} - {LBS}}(M)}}} \\{= {\left( {\frac{L_{{init},{{LL} - {LBS}}}(k)}{a_{LBS}} + N} \right)\left\lbrack {{L_{{init},{{LL} - {LBS}}}(k)} -} \right.}} \\{\left. {k + \frac{M}{2}} \right\rbrack a_{LBS}}\end{matrix} & (160)\end{matrix}$while the prediction-filters method requires:

$\begin{matrix}\begin{matrix}{{L_{{2D},{P - {Filters}}}\left( {k,{N \times M}} \right)} = {\left( {\frac{N}{2^{k}} + \frac{L_{{init},{P - {Filters}}}(k)}{a_{PF}}} \right){L_{P - {Filters}}\left( {k,M} \right)}}} \\{= {\left( {\frac{L_{{init},{P - {Filters}}}(k)}{a_{PF}} + \frac{N}{2^{k}}} \right)\left( {{L_{{init},{P - {Filters}}}(k)} +} \right.}} \\{\left. \frac{M}{2^{k}} \right)a_{PF}}\end{matrix} & (161)\end{matrix}$The results of (160), (161) show that the prediction-filters methodachieves a delay proportional to

$\frac{N \times M}{2^{2k}}a_{PF}$processing cycles for the complete to overcomplete transform derivationof resolution level k for an N×M input frame, while the LL-LBS approachachieves a fixed delay proportional to

${\frac{N \times M}{2}a_{LBS}},$under the same assumptions of system parallelism.

From the above, the skilled person appreciates certain aspects of theinvention as it relates to the complete to overcomplete DWT derivation.For the level-by-level calculation of the overcomplete DWT forbiorthogonal point-symmetric filter-pairs, the requiredcomputational-budget is reduced by creating a structure that exploitsthe symmetries of this new approach. A complexity analysis reveals thatthe proposed prediction-filters scheme in accordance with the inventionis more efficient than the conventional method for the complete toovercomplete transform derivation in a scalable video coding systembecause it offers computational savings and a scalable reduction in thesystem delay. These features lead to inherent computational scalabilityfor the prediction-filters, since, under fixed delay-constraints for theperformance of the complete-to-overcomplete transform for everyresolution level, the codecs that work in resolution-level k can beimplemented with filter-kernels that operate in 2^(2k−2) times smallerclock-frequency than the kernels used in the system that operates in thefull-resolution level (k=1). Conversely, using the same implementationtechnology for all resolution levels, in the low-resolution codecs thelevel of parallelism can be reduced by well-known techniques such asfolding or retiming without surpassing the delay-constraint of thefull-resolution codec. This in turn leads to simple and morepower-efficient. The scalability in the power consumption can be acritical point in the total system design since in many cases very-lowresolution video-codecs are designed for portable devices.

From the flexibility point of view, by thresholding the filter-taps ofthe update structure implementation, the number of multiplicationsrequired for the prediction filters in reduced without a substantialloss in the accuracy of the produced results, something that is not thecase for the LL-LBS method. In addition, the prediction-filters systempossesses a simple implementation structure, since it consists of theparallel application of a set of filters to the same input(single-rate), while the LL-LBS approach requires precise control on thedata-flow in the multirate implementation-structure in order to achievea high-parallel system. As a result, the selective construction ofspecific blocks of the various subbands of the overcomplete transform inthe decoder is expected to be much easier in the prediction-filterssystem than in the conventional LL-LBS approach.

In the following implementations of the above methods are described. Infurther embodiments of the invention a video encoder and/or decoderbased on the in-band motion compensation theory (codec of FIG. 8),either in the spatial or wavelet domain, are disclosed.

In a second embodiment the ‘bottom-up’ ODWT (overcomplete discretewavelet transform), based on the in-band motion compensation methodsdescribed above, can be used in a wavelet video encoder in the followingway (see FIGS. 8 and 9). The wavelet video encoder has the in-bandstructure, which means that motion compensation (MC) performed in themotion compensation module 128 is performed in the wavelet domain. Themotion vectors (MV) 132, provided by a motion estimation (ME) algorithm(e.g., block-based) carried out in the motion estimation module 130, arepreferably given with highest possible accuracy and are either estimatedin the spatial (see embodiment of FIG. 8) or in the wavelet domain (seeembodiment of FIG. 9). The ‘bottom-up’ ODWT module 122 takes as inputthe reconstructed reference wavelet subband image after the inversequantization IQ in IQ module 111 and summation in summer 114. Theovercomplete representation generated by application of a digital filteris stored in a memory buffer (Oref FM) 124. The motion compensationprocess carried out in the motion compensation module 128 takes as inputthe motion vectors and this overcomplete representation. Depending onthe motion vectors, blocks of coefficients are selected using aselection means in the motion compensation unit 128 from theovercomplete representation. These coefficients are combined into awavelet decomposition that is a prediction of the wavelet decompositionof the current image of the video sequence. This prediction issubtracted from the wavelet decomposition of the current image insubtracter 106 resulting in a wavelet subband error image. This subbanderror image is subsequently quantized in quantizer 107 and coded insubband coder 108. The subband error image is also inversely quantizedin IQ module 111 and added to the previous prediction from the MC module128 in adder 114 in order to obtain the next reconstructed referencewavelet subband image to be used by the ‘bottom-up’ ODWT module 122 inthe following iteration.

The main difference between FIGS. 8 and 9 is that in FIG. 8 the motionestimation is carried out in the spatial domain whereas in FIG. 9 thesevectors are determined in the wavelet domain. In the embodiment of FIG.8 the motion vectors are supplied to the motion compensation module andthe receiver from the motion estimation module 130. A selection means inMC module 128 selects the most appropriate subbands, e.g., blocksubbands, from the overcomplete representation in buffer 124. In FIG. 9the motion estimation is made using the subbands of the overcompleterepresentation stored in buffer 124. This motion cestimation can be madelevel-by-level.

The wavelet video decoder (see FIG. 10, having modules common to bothembodiments of FIGS. 8 and 9) operates in an analogous way. The inputfor the subband decoder module 134 is the quantized and coded waveletsubband error image. The decoded error image is supplied to the inversequantizer 135 which ouputs an inversely quantized subband error imagewhich is summated in the summer 138 with the predicted wavelet subbandimage originating from the motion compensation module 142. This resultsin the reconstruction of the current wavelet subband image of the videosequence, which will subsequently be reconstructed by the IDWT (inverseDWT) module 140. The reconstructed wavelet subband image is also theinput for the ‘bottom-up’ ODWT module 148 that generates theovercomplete representation to be stored into the memory buffer (OrefFM), e.g., using application of predictor digital filters as describedabove. Then the motion compensation module 142 selects waveletcoefficients from this overcomplete representation based on the decodedmotion vectors 150. These wavelet coefficients are combined into awavelet decomposition that is a prediction of the current waveletsubband image of the video sequence. This prediction will be summated tothe decoded and inversely quantized wavelet subband error image in thefollowing iteration in summer 138. Note that the switches 116, 118; 152in both the encoders of FIGS. 8 and 9 and the decoder of FIG. 10,respectively provide either the intra (open switch) or the inter (closedswitch) encoding/decoding state. The text above is a description of theinter encoding/decoding state. In the intra encoding/decoding state, afull wavelet subband image is processed instead of a subband errorimage.

Alternatively formulated the proposed coding scheme with motionestimation in the wavelet domain can be applied to the encoder shown inFIG. 1 a. The operation shows a hybrid coding structure; thus whencoding in intra-frame mode, the current frame is wavelet decomposed (DWTmodule 2) and compressed with a block-based intra-band coder (SBC module8). This intra-compressed frame is decompressed (SBD module 12) and thereconstructed wavelet decomposition is used by the CODWT module 22. Thismodule 22 operates in a subband-by-subband manner and constructs fromthe critically-sampled decomposition of every level, the overcompletetransform of that level, which contains all the subband information thatis missing from the critically sampled pyramid due to the subsamplingoperations of every level. An important feature of the invention is thatfor this operation, methods of the invention provide fastcalculation-algorithms with identical algorithmic performance to LBS.These calculations include digital filters. The overcompleterepresentation is used for in-band motion estimation since it is shiftinvariant. The motion compensation is performed in module 28 in thecritically-sampled pyramid, which is subsequently subtracted from thecurrent frame and coded (error frame) in the subtractor 6. The decoderoperates in a similar manner as seen from the lower part of FIG. 1 b.Thus in the proposed structure, there are transform-domain intra-framesand error-frames. The embeddedness of wavelet-based coding and thelevel-by-level operation for the motion estimation and compensationguarantees that drift-free scalability in the decoding can be achieved.To evaluate the performance of the proposed framework, some preliminaryresults are shown in Table IX for the first 48 frames of the footballsequence (luminance component) for two different bit-rates. Thecompression was performed by using one I-frame (intra) and sevenP-frames (inter); the target bit-rates of Table IX for the decompressionwere simply met by decompressing at fixed rates the intra and errorframes. For example, for 760 Kbps, the I-frames were decompressed at 1.0bpp and the P-frames at 0.2 bpp. It can be seen that the proposedalgorithm, even without any sophisticated rate-allocation mechanism, iscompetitive to the 3-D SPIHT algorithm of Kim and Pearlman andoutperforms MPEG-2. FIG. 14 shows an example of frame 5 decompressed atvarious resolutions with the proposed scheme. For all experiments, thesequence was compressed once and all the presented results whereachieved by simply decompressing to various bit-rates and resolutions.The conclusion is that the approach provides resolution, quality andtemporal scalability in video compression with the hybrid-codingstructure if the in-band approach is adopted. The hybrid in-band (motionestimation and compensation in the transform domain) approach usingovercomplete representations equipped with the fast calculation methodscan decode to a variety of bit-rates and resolutions achieving at thesame time very competitive performance to state of the art 3-D waveletvideo-coding and coding standards, which are often proposed in theliterature as the best solution for fine granularity scalability invideo compression.

In a third embodiment wavelet video encoders and decoder based on thein-band motion compensation theory exploiting the in-band motioncompensation theory based prediction rules are disclosed.

Instead of using the ‘bottom-up’ ODWT in the video encoder, the waveletcoefficients required by the motion compensation process can directly becalculated from the reference wavelet subband image without the need forcalculating all of the subbands of an overcomplete representation.Instead, only those subbands are calculated as required, whereby thesesubbands can belong to the set of subbands of the overcompleterepresentation. The propositions in the in-band motion compensationtheory provide the required motion compensating prediction rules thatcorrespond to all possible translations. These prediction rules make useof prediction filters and of particular subbands of the referencewavelet subband image. Which prediction filters to use and whichsubbands to choose, is determined by the motion vectors. If the motionestimation process is block-based, then the prediction rules only takeas input blocks of wavelet coefficients from the reference waveletsubband image. Hence, this ‘bottom-up’ motion compensation process doesnot require the computation of a total overcomplete representation, butcalculates the required wavelet coefficients directly on ablock-by-block basis. Such a wavelet video encoder is shownschematically in FIG. 11 and the decoder in FIG. 12. The majordifference between the embodiment of FIG. 11 and that of FIG. 9 is thatthe motion compensation module 128 uses the motion vector informationobtained from the motion estimation module 130 to calculate only thebest subbands for motion compensation. The method of calculation relieson digital predictor filters as described above but avoids therequirement to calculate all the subbands of the overcompleterepresentation. In FIG. 11 motion estimation is performed in the waveletdomain, but it can also be performed in the spatial domain as in FIG. 8.

In a fourth embodiment resolution scalable wavelet video encoders anddecoders based on the in-band motion compensation theory are disclosed.

The wavelet video codecs, disclosed in the first to third embodiments,are both capable of supporting resolution scalability while avoiding adrift between the encoder and the decoders of different resolutionlevels. Conversely, a spatial domain motion compensation encoder causesa drift between what is encoded at full resolution and what is beingdecoded at a lower resolution. The reason for this drift is that theencoder uses all information available in the full resolution image (allsubbands), which is not all transmitted to the decoders. A solutionwould be to code every resolution level independently from each other sothat all possible configurations of decoders are supported, i.e.multicasting. Multicasting is however very inefficient from the encoderspoint of view, because multiple encoded streams have to be generated.Also from a compression point of view, the efficiency is very low and alot of redundancy between resolution levels remains. This redundancy canpartially be removed by working in a layered fashion, which is theMPEG-4 and H.263+approach. First a base layer, with lowest resolution,is encoded with a certain image quality. The result is subtracted frameby frame (after upsampling and interpolating) from a version of thevideo sequence with twice the resolution of the base layer. Theresulting error frames form an enhancement layer and are coded (with orwithout motion compensation). Several enhancement layers can begenerated in this way. Although this principle is an improvement ofmulticasting, a further improvement can be obtained by a hierarchicalcoding approach. This approach requires a ‘hierarchical’ transform likethe wavelet transform. The advantage of the wavelet transform is that itis critically sampled and supports multiple resolution scales. This isin contrast to the layered approach, where the total number of coded‘coefficients’ is higher than the number of pixels of the fullresolution image. Hence, an improved coding efficiency can be expectedfrom the hierarchical coding approach. However, motion compensationshould be performed in the transform domain to make full advantage ofthe hierarchical resolution property of the wavelet transform for videocoding, hence the wavelet video codec structures of FIGS. 8 and 11,respectively are required. In order to exploit the resolutionscalability property of the wavelet transform in the second and thirdembodiment coders, a simple precaution needs to be taken. Whenperforming motion compensation on subbands (or subband blocks for thethird embodiment coder) that are required to reconstruct a certainresolution level, subbands of lower wavelet levels should not be used inthe motion compensation process (lower wavelet level corresponds tohigher resolution level). For the ‘bottom-up’ ODWT this means thatsubbands of lower wavelet levels than the level of the subbands of whichan overcomplete representation is generated, should not be used. Thereason for this is that a decoder for this particular resolution level,does not receive the information that corresponds to the subbands of thelower wavelet levels and hence can not use it. In practice this meansthat the propositions or prediction rules of the in-band motioncompensation theory should be adjusted. Terms in these prediction rules,which correspond to lower wavelet level subbands, should be neglected toenable resolution scalability by the wavelet video encoders of the firsttwo embodiments.

In a fifth embodiment a method for wavelet-based in-band motioncompensation and a digital, being useful for the first four embodimentsis disclosed.

A novel in-band motion compensation method for wavelet-based videocoding is proposed, derived here for the 1-D discrete wavelet transformbut not limited thereto. The fundamental idea is the computation of therelationships between the subbands of the non-shifted input signal andthe subbands corresponding to all translations of the input signal. Themotion compensation method is formalized into prediction rules.

The in-band motion compensation algorithm can be extended to 2-D images,decomposed using the separable 2-D WT. The chosen approach is valid forN-level decompositions. The prediction rules of the algorithm allow thein-band motion compensation process to reach a zero prediction error forall translations of the image pixels. Besides the applicability of this2-D in-band motion compensation algorithm for scalable video codecs,other applications are envisaged such as in-band motion estimation,overcomplete wavelet decompositions, etc.

The discrete wavelet transform (WT) is inherently scalable in resolutionand quality, which is a very much wanted property for streaming videoe.g., on the Internet. A possible codec structure exploiting the WT forvideo coding is the in-band codec. In-band means that motioncompensation is performed on the wavelet coefficients. However, theperiodic translation-invariance of the critically sampled WT prohibitsfrom reaching a zero prediction error on wavelet decomposition level J,if the shift or translation r of the image pixels does not belong toτ=2^(J) k, with kεZ. In the following, we suppose that the translationof the image pixels is known. The novel motion compensation algorithm ormethod will be derived for the 1-D WT.

Concerning the concept of periodic translation invariance we denote byY(z) the result of filtering the input signal X(z) with filter H(z).Y(z) subsampled with factor M=2 is not invariant with respect to integershifts or translations τεZ of X(z). This is caused by the subsamplingoperator that retains, depending on τ, either a shifted version of theeven or the odd indexed samples of Y(z), which we denote by Y_(even)(z)and Y_(odd)(z). In general, only a limited number of possible delayedoutputs exist, equal to the subsampling factor M. Therefore, this systemis called periodic translation-invariant. Given an even translation τ=2k with kεZ, we obtain the following expression for the shifted, filteredand subsampled signal Y^(2k)(z):Y ^(2k)(z)=z ^(k) [H ₀(z)X ₀(z)+H ₁(z)X ₁(z)]=z ^(k) Y_(even)(z),  (186)with X₀(z), X₁(z) the polyphase components of X(z) and H₀(z), H₁(z) theType I polyphase components of H(z) [20]. This actually istranslation-invariance, because the output sequence y^(2k)(n) is ashifted version of sequence y^(even)(n), obtained for τ=0. Conversely,for an odd translation τ=2k+1 the output signal is given by:Y ^(2k+1)(z)=z ^(k) [H ₁(z)X ₀(z)+zH ₀(z)X ₁(z)]=z ^(k) Y_(odd)(z)  (187)

Now in-band motion compensation for 1-D WT, Level 1 is derived.Depending on the translation r of the input signal, we obtain a shiftedversion of either Y_(even)(z) or Y_(odd)(z). Hence, motion compensationcan be simplified to prediction of odd samples from even samples or viceversa. For the single level WT with filters H(z) and G(z), we denoterespectively the average subband by A_(even)(z), and the detail subbandby D_(even)(z). We obtain the output signals z^(k) A_(even)(z), z^(k)D_(even)(z) for τ=2k, and z^(k) A_(odd)(z) and z^(k) D_(odd)(z) forτ=2k+1. FIG. 13 (level 1) represents a decomposition tree containing thesubbands of the single level WT for different translations τ. Withoutany loss of generality one can drop the delays z^(k), and one can reducethe in-band motion compensation problem to predicting the subbandsA_(odd)(z) and D_(odd)(z) from the reference subbands A_(even)(z) andD_(even)(z). This results in the first prediction rule:

-   -   Translation τ=2k+1, level 1:        D _(odd)(z)=P ₀(z)A _(even)(z)+P ₁(z)D _(even)(z), with  (188)        P ₀(z)=D ⁻¹ [G ₁(z)G ₁(z)−zG ₀(z)G ₀(z)], P ₁(z)=D ⁻¹ [zH ₀(z)G        ₀(z)−H ₁(z)G ₁(z)].        A _(odd)(z)=P ₂(z)A _(even)(z)+P ₃(z)D _(even)(z), with  (189)        P ₂(z)=D ⁻¹ [H ₁(z)G ₁(z)−zH ₀(z)G ₀(z)], P ₃(z)=D ⁻¹ [zH ₀(z)H        ₀(z)−H ₁(z)H ₁(z)].        Proof: From the system of equations formed by the expressions        for A_(even)(z) and D_(even)(z) in polyphase notation (186), we        can isolate the input signal polyphase components X₀(z) and        X₁(z):

$\begin{matrix}{{{X_{0}(z)} = {\frac{1}{{DetH}_{\underset{\_}{p}}(z)}\left\lbrack {{{G_{1}(z)}{A_{even}(z)}} - {{H_{1}(z)}{D_{even}(z)}}} \right\rbrack}},} & (190) \\{{{X_{1}(z)} = {\frac{1}{{DetH}_{\underset{\_}{p}}(z)}\left\lbrack {{{H_{0}(z)}{D_{even}(z)}} - {{G_{0}(z)}{A_{even}(z)}}} \right\rbrack}},} & (191)\end{matrix}$with H_(p) (z) the analysis polyphase matrix [20]. Because the waveletfilters satisfy the perfect reconstruction condition, we necessarilyhave DetH_(p) (z)=Dz^(−q) [21], with D and q constants. To simplify theformulas, we consider q=0. If we substitute the signal polyphasecomponents into the polyphase expression (187) for D_(odd)(z) we obtain(188). Similarly, if we substitute (190) and (191) into the expression(187) for A_(odd)(Z) we obtain (189).

Similarly to the previous calculations, we have to determine theinfluence of the X(z) translation on the subbands of level 2. Considerthe decomposition either of the subband z^(k) A_(even)(z) or z^(k)A_(odd)(z) as illustrated in FIG. 13. This is equivalent to a singlelevel wavelet decomposition applied to the signals A_(even)(z) andA_(odd)(z) which are shifted either by an even or odd translation k, orequivalently k=2l and k=2l+1 with εZ. As a result, we obtain fouralternative decompositions corresponding to the following translationsτ: τ=4l, τ=4l+2, τ=4l+1 and τ=4l+3, see FIG. 13.

From a motion compensation point of view, the invention predicts for allfour translation types, the decomposition subbands starting from thereference subbands of the 2-levels WT, corresponding to τ=0:D_(even)(z), A_(even) ^(A) ^(even) (z) and D_(even) ^(A) ^(even) (z).The upper index indicates the decomposed subband of the lower level. Ifthe translation is of type τ=4l, then we have translation-invariance.All other translation types r belong to the translation-variantcategory. The following prediction rules can be formulated:

-   -   Translation τ=4l+2, level 2:        Apply prediction rule for τ=2k+1 to the subbands A_(even) ^(A)        ^(even) (z), and D_(even) ^(A) ^(even) (z).    -   Translation τ=4l+1, level 2:

$\begin{matrix}{{{{A_{even}^{A_{odd}}(z)} = {{{Q_{0}(z)}{A_{even}^{A_{even}}(z)}} + {{Q_{1}(z)}{D_{even}^{A_{even}}(z)}} + {A_{even}^{{P_{3}{(z)}}{D_{even}{(z)}}}(z)}}},{with}}{{{Q_{0}(z)} = {{P_{2,0}(z)} - {z^{- 1}{P_{1}(z)}{P_{2,1}(z)}}}},\mspace{50mu}{{Q_{1}(z)} = {z^{- 1}{P_{3}(z)}{P_{2,1}(z)}}},}} & (192)\end{matrix}$

$\begin{matrix}{{{{D_{even}^{A_{odd}}(z)} = {{{Q_{2}(z)}{A_{even}^{A_{even}}(z)}} + {{Q_{3}(z)}{D_{even}^{A_{even}}(z)}} + {D_{even}^{{P_{3}{(z)}}{D_{even}{(z)}}}(z)}}},{with}}{{{Q_{2}(z)} = {z^{- 1}{P_{0}(z)}{P_{2,1}(z)}}},\mspace{50mu}{{Q_{3}(z)} = {{P_{2,0}(z)} + {z^{- 1}{P_{1}(z)}{{P_{2,1}(z)}.}}}}}} & (193)\end{matrix}$P_(2,0)(z) and P_(2,1)(z) are the polyphase components of

P₂(z);(A/D)_(even)^( _(P₃(z)D_(even^((z)))))(z)is the filtered and subsampled signal, originating from P₃(z)D_(even)(z)by respectively filtering with H(z) or G(z), and retaining the evensamples.Translation τ=4l+3: analogous to τ=4l+1.

The level 3 part of the 1-D WT in-band compensation rules is nowdiscussed. The following classification of translation parameter τ onlevel 3 is found: τ=8m, τ=8m+4, τ=8m+2, τ=8m+6, τ=8m+1, τ=8m+5, τ=8m+3and τ=8m+7 (FIG. 13). For the left branch of the decomposition tree ofFIG. 13, i.e. the branch corresponding to even translations τ=2k, we canapply the prediction rules from the previous section. However, for theright branch corresponding to odd translations τ=2k+1, we describe thein-band motion compensation relationships through new prediction rules.We formulate the prediction rule for shift τ=8m+1. We refer to detaileddescription for the other prediction rules and the proofs.

-   -   Translation τ=8m+1, level 3:

$\begin{matrix}{{{A_{even}^{A_{even}^{A_{odd}}}(z)} = {{{R_{0}(z)}{A_{even}^{A_{even}^{A_{even}}}(z)}} + {{R_{1}(z)}{D_{even}^{A_{even}^{A_{even}}}(z)}} + {z^{- 1}A_{odd}^{z\;{Q_{1}{(z)}}{D_{even}^{A_{even}}{(z)}}}} + A_{even}^{A_{even}^{P_{3{(z)}D_{{even}{(z)}}}}}}},} & (194) \\{{{R_{0}(z)} = {{Q_{0,0}(z)} - {z^{- 1}{P_{1}(z)}{Q_{0,1}(z)}}}},\mspace{104mu}{{R_{1}(z)} = {z^{- 1}{P_{3}(z)}{Q_{0,1}(z)}}},} & \; \\{{{D_{even}^{A_{even}^{A_{odd}}}(z)} = {{{R_{2}(z)}{A_{even}^{A_{even}^{A_{even}}}(z)}} + {{R_{3}(z)}{D_{even}^{A_{even}^{A_{even}}}(z)}} + {z^{- 1}D_{odd}^{z\;{Q_{1}{(z)}}{D_{even}^{A_{even}}{(z)}}}} + D_{even}^{A_{even}^{P_{3{(z)}D_{{even}{(z)}}}}}}},} & (195) \\\; & \; \\{{{R_{2}(z)} = {z^{- 1}{P_{0}(z)}{Q_{0,1}(z)}}},\mspace{169mu}{{R_{3}(z)} = {{Q_{0,0}(z)} + {z^{- 1}{P_{1}(z)}{{Q_{0,1}(z)} \cdot}}}}} & \;\end{matrix}$

The same approach can be followed as in the previous paragraphs toobtain prediction rules for higher decomposition levels. Furthermore, arecursive pattern is revealed when deriving the R-prediction filtersfrom the Q-prediction filters. This allows us to deduce the predictionfilters for the fourth level and for higher levels without relying onany calculations.

FIG. 16 shows the implementation of a coder/decoder which can be usedwith the invention implemented using a microprocessor 230 such as aPentium IV from Intel Corp. USA. The microprocessor 230 may have anoptional element such as a co-processor 224, e.g., for arithmeticoperations or microprocessor 230-224 may be a bit-sliced processor. ARAM memory 222 may be provided, e.g., DRAM. Various I/O (input/output)interfaces 225, 226, 227 may be provided, e.g., UART, USB, I²C businterface as well as an I/O selector 228. FIFO buffers 232 may be usedto decouple the processor 230 from data transfer through theseinterfaces. A keyboard and mouse interface 234 will usually be providedas well as a visual display unit interface 236. Access to an externalmemory such as a disk drive may be provided via an external businterface 238 with address, data and control busses. The various blocksof the circuit are linked by suitable busses 231. The interface to thechannel is provided by block 242 which can handle the encoded videoframes as well as transmitting to and receiving from the channel.Encoded data received by block 242 is passed to the processor 230 forprocessing.

Alternatively, this circuit may be constructed as a VLSI chip around anembedded microprocessor 230 such as an ARM7TDMI core designed by ARMLtd., UK which may be synthesized onto a single chip with the othercomponents shown. A zero wait state SRAM memory 222 may be providedon-chip as well as a cache memory 224. Various I/O (input/output)interfaces 225, 226, 227 may be provided, e.g., UART, USB, I²C businterface as well as an I/O selector 228. FIFO buffers 232 may be usedto decouple the processor 230 from data transfer through theseinterfaces. A counter/timer block 234 may be provided as well as aninterrupt controller 236. Access to an external memory may be providedan external bus interface 238 with address, data and control busses. Thevarious blocks of the circuit are linked by suitable busses 231. Theinterface to the channel is provided by block 242 which can handle theencoded video frames as well as transmitting to and receiving from thechannel. Encoded data received by block 242 is passed to the processor230 for processing.

Software programs may be stored in an internal ROM (read only memory)246 which may include software programs for carrying out subbanddecoding and/or encoding in accordance with any of the methods of theinvention. In particular software programs may be provided for digitalfilters according to embodiments of the invention described above to beapplied to a reference or other frame of data to generate one or moresubbands of a set of subbands of an overcomplete representation of theframe by calculations at single rate. That is the software, whenexecuted on the processor 230 carries out the function of any of themodules 22, 122, 48, 148, 128, 142 described above. Software may alsocombine the functions of several modules, e.g., the module 30 and module22 of FIG. 1 a or the module 130 and module 122 of FIG. 9, or acombination of the modules 22, 28 and 30 of FIG. 1 a or modules 122,128, 130 of FIGS. 8, 9 or 11 or modules 122 and 128 of FIG. 8 or modules128 and 130 of FIG. 11 or modules 142 and 148 of FIG. 10. The methodsdescribed above may be written as computer programs in a suitablecomputer language such as C and then compiled for the specific processorin the design. For example, for the embedded ARM core VLSI describedabove the software may be written in C and then compiled using the ARM Ccompiler and the ARM assembler. Reference is made to “ARMSystem-on-chip”, S. Furber, Addison-Wiley, 2000. The invention alsoincludes a data carrier on which is stored executable code segments,which when executed on a processor such as 230 will execute any of themethods of the invention, in particular will execute digital filteringaccording to embodiments of the invention described above to be appliedto a reference or other frame of data to generate one or more subbandsof a set of subbands of an overcomplete representation of the frame bycalculations at single rate. The data carrier may be any suitable datacarrier such as diskettes (“floopy disks”), optical storage media suchas CD-ROMs, DVD ROM's, tape drives, hard drives, etc. which are computerreadable.

FIG. 17 shows the implementation of a coder/decoder which can be usedwith the invention implemented using an dedicated filter module.Reference numbers in FIG. 17 which are the same as the reference numbersin FIG. 16 refer to the same components—both in the microprocessor andthe embedded core embodiments.

Only the major differences will be described with respect to FIG. 17.Instead of the microprocessor 230 carrying out methods required tosubband encode and decode a bitstream this work is now taken over by asubband coding module 240. Module 240 may be constructed as anaccelerator card for insertion in a personal computer. The subbandmodule has means for carrying out subband decoding and/or encoding inaccordance with any of the methods of the invention. In particular, themodule may be provided with means for digital filtering according toembodiments of the invention described above to be applied to areference or other frame of data to generate one or more subbands of aset of subbands of an overcomplete representation of the frame bycalculations at single rate. These filters may be implemented as aseparate filter module 241, e.g., an ASIC (Application SpecificIntegrated Circuit) or an FPGA (Field Programmable Gate Array) havingmeans for digital filtering according to embodiments of the inventiondescribed above to be applied to a reference or other frame of data togenerate one or more subbands of a set of subbands of an overcompleterepresentation of the frame by calculations at single rate.

Similarly, if an embedded core is used such as an ARM processor core oran FPGA, a subband coding module 240 may be used which may beconstructed as a separate module in a multi-chip module (MCM), forexample or combined with the other elements of the circuit on a VLSI.The subband module 240 has means for carrying out subband decodingand/or encoding in accordance with any of the methods of the invention.In particular, the module may be provided with means for digitalfiltering according to embodiments of the invention described above tobe applied to a reference or other frame of data to generate one or moresubbands of a set of subbands of an overcomplete representation of theframe by calculations at single rate. As above, these filters may beimplemented as a separate filter module 241, e.g., an ASIC (ApplicationSpecific Integrated Circuit) or an FPGA (Field Programmable Gate Array)having means for digital filtering according to embodiments of theinvention described above. The invention also includes other integratedcircuits such as ASIC's or FPGA's which carry out the function of any ofthe modules 22, 122, 48, 148, 128, 142 described above. Such integratedcircuits may also combine the functions of several modules, e.g., themodule 30 and module 22 of FIG. 1 a or the module 130 and module 122 ofFIG. 9, or a combination of the modules 22, 28 and 30 of FIG. 1 a ormodules 122, 128, 130 of FIGS. 8, 9 or 11 or modules 122 and 128 of FIG.8 or modules 128 and 130 of FIG. 11 or modules 142 and 148 of FIG. 10.

While the invention has been shown and described with reference tocertain embodiments, it will be understood by those skilled in the artthat various changes or modifications in form and detail may be madewithout departing from the scope and spirit of this invention.

TABLES

TABLE I The 9/7 filter-bank modified so that Det = H₀G₁ − G₀H₁ = −1. Inthis case G is centered around zero (linear phase). n h(n) M g(m) 1  0.85269867900889 0   0.78848561640558 0, 2   0.37740285561283 −1, 1  −0.41809227322162 −1, 3   −0.11062440441844 −2, 2   −0.04068941760916−2, 4   −0.02384946501956 −3, 3     0.06453888262870 −3, 5    0.03782845550726 −4, 4  

TABLE II The prediction filters of level 1 for the 9/7 filter pair ofTable I. Degree in Z F₀ ¹ F₁ ¹ F₂ ¹ F₃ ¹ 4 −0.00244140625001  0.00143099204607 −0.00416526737096   0.00244140625001 3  0.02392578125006 −0.00893829770284   0.05562204500420−0.02392578125006 2 −0.11962890624961   0.09475201933935−0.18500077367828   0.11962890624961 1   0.59814453124955−0.32145927076690   0.26708799209208 −0.59814453124955 0  0.59814453124955   0.46842911416863 −0.18500077367828−0.59814453124955 −1   −0.11962890624961 −0.32145927076690  0.05562204500420   0.11962890624961 −2     0.02392578125006  0.09475201933935 −0.00416526737096 −0.02392578125006 −3    0.00244140625001 −0.00893829770284   0.00244140625001 −4    0.00143099204607

TABLE III The prediction filters of level 2 for the 9/7 filter pair ofTable I. Degree in Z F₀ ² F₁ ² F₂ ² F₃ ² 5 −0.00005841255188  0.00003423760266 −0.00009965727597   0.00005841255188 4−0.00088787078858   0.00064208431103 −0.00116063101768  0.00088787078858 3   0.01174092292788 −0.00325056581524  0.02934202037389 −0.01174092292788 2 −0.06254196166961  0.05005002314451 −0.11091074747671   0.05765914916960 1  0.26671171188334 −0.19238483073456   0.17732657799233−0.50596952438255 0   0.88179588317802   0.31072164151829−0.14082618249241   0.31449317932109 −1   −0.12007284164365−0.24526493865167   0.05464973467748   0.16792440414377 −2    0.02710342407219   0.09377374163572 −0.00869377426124−0.02710342407219 −3   −0.00403046607971 −0.01586242405270  0.00036249037151   0.00403046607971 −4     0.00023365020752  0.00169389067232   0.00001016910979 −0.00023365020752 −5    0.00000596046448 −0.00014936599745 −0.00000596046448 −6  −0.00000349363292 Degree in Z F₄ ² F₅ ² F₆ ² F₇ ² 6   0.00000596046448−0.00000349363292   0.00001016910979 −0.00000596046448 5  0.00023365020752 −0.00014936599745   0.00036249037151−0.00023365020752 4 −0.00403046607971   0.00169389067232−0.00869377426124   0.00403046607971 3   0.02710342407219−0.01586242405270   0.05464973467748 −0.02710342407219 2−0.12007284164365   0.09377374163572 −0.14082618249241  0.16792440414377 1   0.88179588317802 −0.24526493865167  0.17732657799233   0.31449317932109 0   0.26671171188334  0.31072164151829 −0.11091074747671 −0.50596952438255 −1  −0.06254196166961 −0.19238483073456   0.02934202037389  0.05765914916960 −2     0.01174092292788   0.05005002314451−0.00116063101768 −0.01174092292788 −3   −0.00088787078858−0.00325056581524 −0.00009965727597   0.00088787078858 −4  −0.00005841255188   0.00064208431103   0.00005841255188 −5    0.00003423760266

TABLE IV The prediction filters of level 3 for the 9/7 filter pair ofTable I. Degree in Z F₀ ³ F₁ ³ F₂ ³ F₃ ³ 6   0.00000014260877−0.00000008358790   0.00000024330390 −0.00000014260877 5−0.00003006192856   0.00001732327610 −0.00005215310877  0.00003006192856 4 −0.00036325305700   0.00027118376662−0.00044706508018   0.00036325305700 3   0.00523493974471−0.00142451855033   0.01314750521441 −0.00523493974471 2−0.02866946801063   0.02253736185968 −0.05285711618290  0.02689372643348 1   0.11828768299888 −0.09155949520252  0.09104952775264 −0.24337160633810 0   0.96858347300390  0.15937234774348 −0.08001261890323   0.79500829335214 −1  −0.07875438081051 −0.13957337353587   0.03592212357721  0.13296122895490 −2     0.01854321861163   0.06187507024177−0.00704239128113 −0.01807591819659 −3   −0.00303826271556−0.01244586898594   0.00027485003322   0.00303826271556 −4    0.00019600149244   0.00107109829644   0.00001711950208−0.00019600149244 −5   −0.00000998261385 −0.00013523300959−0.00000002482693 −0.00000998261385 −6   −0.00000001455192−0.00000582084131   0.00000001455192 −7     0.00000000852938 Degree in ZF₄ ³ F₅ ³ F₆ ³ F₇ ³ 6   0.00000216765329 −0.00000127053604  0.00000369821923 −0.00000216765329 5   0.00013144733384−0.00008156099626   0.00021111880330 −0.00013144733384 4−0.00354297226295   0.00173673401727 −0.00698737064387  0.00354297226295 3   0.02792382379991 −0.01348353219813  0.06026756896080 −0.02804064890367 2 −0.13103966135465  0.10299880531837 −0.17816629613736   0.15452150721042 1  0.75361341703605 −0.30990650725195   0.24203685966003−0.22018999326938 0   0.43101294059278   0.42438892216659−0.15941216055657 −0.67115862388008 −1   −0.09466141927950−0.27675857877612   0.04435604503300   0.08660048712007 −2    0.01815370982519   0.07550649033473 −0.00220859321570−0.01814178889623 −3   −0.00153230270371 −0.00547825603015−0.00009989690557   0.00153230270371 −4   −0.00006058020517  0.00104172325094 −0.00000097321559   0.00006058020517 −5  −0.00000057043508   0.00003669634916   0.00000057043508 −6    0.00000033435159 Degree in Z F₈ ³ F₉ ³ F₁₀ ³ F₁₁ ³ 6 −0.00000057043508  0.00000033435159 −0.00000097321559   0.00000057043508 5−0.00006058020517   0.00003669634916 −0.00009989690557  0.00006058020517 4 −0.00153230270371   0.00104172325094−0.00220859321570   0.00153230270371 3   0.01815370982519−0.00547825603015   0.04435604503300 −0.01814178889623 2−0.09466141927950   0.07550649033473 −0.15941216055657  0.08660048712007 1   0.43101294059278 −0.27675857877612  0.24203685966003 −0.67115862388008 0   0.75361341703605  0.42438892216659 −0.17816629613736 −0.22018999326938 −1  −0.13103966135465 −0.30990650725195   0.06026756896080  0.15452150721042 −2     0.02792382379991   0.10299880531837−0.00698737064387 −0.02804064890367 −3   −0.00354297226295−0.01348353219813   0.00021111880330   0.00354297226295 −4    0.00013144733384   0.00173673401727   0.00000369821923−0.00013144733384 −5     0.00000216765329 −0.00008156099626−0.00000216765329 −6   −0.00000127053604 Degree in Z F₁₂ ³ F₁₃ ³ F₁₄ ³F₁₅ ³ 7 −0.00000001455192   0.00000000852938 −0.00000002482693  0.00000001455192 6   0.00000998261385 −0.00000582084131  0.00001711950208 −0.00000998261385 5   0.00019600149244−0.00013523300959   0.00027485003322 −0.00019600149244 4−0.00303826271556   0.00107109829644 −0.00704239128113  0.00303826271556 3   0.01854321861163 −0.01244586898594  0.03592212357721 −0.01807591819659 2 −0.07875438081051  0.06187507024177 −0.08001261890323   0.13296122895490 1  0.96858347300390 −0.13957337353587   0.09104952775264  0.79500829335214 0   0.11828768299888   0.15937234774348−0.05285711618290 −0.24337160633810 −1   −0.02866946801063−0.09155949520252   0.01314750521441   0.02689372643348 −2    0.00523493974471   0.02253736185968 −0.00044706508018−0.00523493974471 −3   −0.00036325305700 −0.00142451855033  0.00005215310877   0.00036325305700 −4   −0.00003006192856  0.00027118376662   0.00000024330390   0.00003006192856 −5    0.00000014260877   0.00001732327610 −0.00000014260877 −6  −0.00000008358790

TABLE V $\begin{matrix}{{The}\mspace{14mu}{multiplications}\mspace{14mu}{required}\mspace{14mu}{for}\mspace{14mu}{the}\mspace{14mu}{production}\mspace{14mu}{of}\mspace{14mu}{the}\mspace{14mu}{subbands}\mspace{14mu}{of}} \\{{{level}\mspace{14mu} k\mspace{14mu}\left( {{{the}\mspace{20mu}{filters}\mspace{20mu}{for}\mspace{14mu}{subbands}\mspace{14mu} A_{1}^{k}},{D_{1}^{k}\mspace{14mu}{are}\mspace{14mu}{omitted}}} \right)},{with}} \\{{l \in {\left\lbrack {2,k} \right\rbrack\mspace{14mu}{and}\mspace{14mu} i} \in {{\left\lbrack {0,{2^{l - 2} - 1}} \right\rbrack.\mspace{14mu}{At}}\mspace{14mu}{level}\mspace{14mu} k}},{subbands}} \\{A_{0}^{k},{D_{0}^{k}\mspace{14mu}{contain}\mspace{14mu}\frac{N}{2^{k}}\mspace{14mu}{{coefficients}.}}}\end{matrix}\quad$ Equation Multiplications Number Convolutions forlevel k  (9) $\begin{matrix}F_{{4i},0}^{l - 1} \\F_{{4i},1}^{l - 1}\end{matrix}\quad$$\frac{N}{2^{k}}\left( {T_{F_{{4i},0}^{l - 1}} + T_{F_{{4i},1}^{l - 1}}} \right)$(10) F_(4i, 1)^(l − 1)[n] * [F₁^(l)[n] * D₀^(k)[n]]$\frac{N}{2^{k}}T_{F_{{4i},1}^{l - 1}}$ (11) F_(4i, 1)^(l − 1)  $\frac{N}{2^{k}}T_{F_{{4i},1}^{l - 1}}$ (12) $\begin{matrix}{{{F_{{4i},0}^{l - 1}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}},} \\{{F_{{4i},1}^{l - 1}\lbrack n\rbrack}*\left\lbrack {{F_{3}^{l}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}} \right\rbrack}\end{matrix}\quad$$\frac{N}{2^{k}}\left( {T_{F_{{4i},0}^{l - 1}} + T_{F_{{4i},1}^{l - 1}}} \right)$(13) ${\begin{matrix}F_{{4i},1}^{l - 1} \\F_{{4i},0}^{l - 1}\end{matrix}\quad}\quad$$\frac{N}{2^{k}}\left( {T_{F_{{4i},0}^{l - 1}} + T_{F_{{4i},1}^{l - 1}}} \right)$(14) F_(4i, 0)^(l − 1)[n] * [F₁^(l)[n] * D₀^(k)[n]]$\frac{N}{2^{k}}T_{F_{{4i},0}^{l - 1}}$ (15) F_(4i, 0)^(l − 1)  $\frac{N}{2^{k}}T_{F_{{4i},0}^{l - 1}}$ (16) ${\begin{matrix}{{{F_{{4i},1}^{l - 1}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}},} \\{{F_{{4i},0}^{l - 1}\lbrack n\rbrack}*\left\lbrack {{F_{3}^{l}\lbrack n\rbrack}*{D_{0}^{k}\lbrack n\rbrack}} \right\rbrack}\end{matrix}\quad}\quad$$\frac{N}{2^{k}}\left( {T_{F_{{4i},0}^{l - 1}} + T_{F_{{4i},1}^{l - 1}}} \right)$

TABLE VI Multiplication budget for the level-by-level production of theovercomplete representation. Two modes of operation are presented:FO-mode and HFO-mode. The results are presented per decomposition levelboth for the original and thresholded filters. Full (9/7 filter-pair)Multipli- Multipli- Reduction Over- Number of cations for cations incomplete taps of prediction prediction- for multipli- (FO) filters:{F_(4i) ^(l−1) (F_(4i) ^(l−1))}, filters LL-LBS cations mode iε[0,2^(l−2)],iεz method method (%) Level {7 (7)}₀  10.25 N   13.5 N 24.1 l =2  (10.25 N) (24.1) Level {11 (7)}₀, {11 (7)}₁   13.375 N  18.25 N 26.7l = 3  (10.375 N) (43.2) Level {13 (7)}₀, (12 (8)}₁, 16.0625 N  23.125 N30.5 l = 4 {12 (8)}₂, {13 (7)}₃  (11.9375 N) (48.4) High- (9/7filter-pair) Frequency Number of taps Multipli- Multipli- ReductionOver- of prediction filters: cations for cations in complete {[F_(4i,0)^(l−1) (F_(4i,0) ^(l−1))], prediction- for multipli- (HFO) [F_(4i,1)^(l−1) (F_(4i,1) ^(l−1))}, filters LL-LBS cations mode iε[0,2^(l−2)],iεz method method (%) Level {[4 (4)], [3 (3)]}₀      5 N   9.75N 48.7 l = 2     (5 N) (48.7) Level {[5 (3)], [6 (4)]}₀,  6.625 N 13.875 N 52.3 l = 3 {[6 (4)], [5 (3)]}₁   (5.125 N) (63.1) Level {[7(3)], [6 (4)]}₀,     8 N 18.4375 N 56.6 l = 4 {[6 (4)], [6 (4)]}₁, (5.9375 N) (67.8) {[6 (4)], [6 (4)]}₂, {[6 (4)], [7 (3)]}₃  Note: Innotation F (F), (F) is the number of filter-taps of filter F, withmagnitude higher than the thresholds shown in Table VIII for thecorresponding level l.Table VI. Miltiplacation budget for the level-by-level production of theovercomplete representation. Two modes of operation are presented:FO-mode and HFO-mode.The results are presented per decomposition levelboth for the original and threshold filters. Note: In notation F (F),(F) is the number of filter taps of filter F, with magnatude higher thanthe thresholds shown in Table VIII for the corresponding level l.

TABLE VII Comparison of the prediction-filters method with the LL-LBSmethod with respect to latency, delay and number of filter-kernels(F-Ks) used. A typical case of a 512-sample signal decomposed with the9/7 filter-pair is presented. The numbers in parenthesis next to thefilter-kernels show the maximum number of taps of these units (takinginto account the filter symmetries) for every decomposition level.Decomposition Prediction-filters LL-LBS level (k) L_(init,P-Filters)L_(P-Filters) F-Ks L_(init,LL-LBS) L_(LL-LBS) F-KsL_(LL-LBS)/L_(P-Filters) 2 8a_(PF) 136a_(PF)  10 (5)  8a_(LBS)264a_(LBS) 10 (5) $1.94_{\frac{\sigma_{LBS}}{\sigma_{PF}}}$ 3 9a_(PF)73a_(PF) 22 (6) 13a_(LBS) 269a_(LBS) 22 (5)$3.68_{\frac{\sigma_{LBS}}{\sigma_{PF}}}$ 4 11a_(PF)  43a_(PF) 46 (7)18a_(LBS) 274a_(LBS) 46 (5) $6.37_{\frac{\sigma_{LBS}}{\sigma_{PF}}}$

TABLE VIII The maximum mean-square error (MMSE) per decomposition levelbetween the results of the original and the thresholded prediction-filters for the grayscale images of the JPEG-2000 test-set. The filtersare applied in the row-column manner and the thresholds refer to thefilters of the update-structure implementation of every level.Decomposition level Threshold value MMSE 2 10⁻³ 0.0   3 10⁻⁴ 0.03315 410⁻⁵ 0.12034

TABLE IX Average PSNR for the first 48 frames of the football sequenceat 30 frames per second. 760 Kbps 29.2 27.9 27.3 26.9 2.54 Mbps 34.034.2 33.5 33.0 *The bit-rate for the motion vectors is not included

APPENDIX I

For the proposition P(1) (equation (1)), the calculated subbands of thefirst decomposition level, namely subbands A₀ ¹,D₀ ¹ and A₁ ¹,D₁ ¹, areshown pictorially in FIG. 2. These subbands can be written using X₀,X₁(the polyphase components of X) and the Type I polyphase components ofH, G] as:A ₀ ¹ =H ₀ X ₀ +H ₁ X ₁,  (162)D ₀ ¹ =G ₀ X ₀ +G ₁ X ₁,  (163)andA ₁ ¹ =H ₁ X ₀ +zH ₀ X ₁,  (164)D ₁ ¹ =G ₁ X ₀ +zG ₀ X ₁.  (165)By solving the system of (162), (163) for the polyphase components X₀,X₁we derive:

$\begin{matrix}{{X_{0} = {\frac{1}{\left( {{H_{0}G_{1}} - {G_{0}H_{1}}} \right)}\left\lbrack {{G_{1}A_{0}^{1}} - {H_{1}D_{0}^{1}}} \right\rbrack}},} & (166) \\{{X_{1} = {\frac{1}{\left( {{H_{0}G_{1}} - {G_{0}H_{1}}} \right)}\left\lbrack {{{- G_{0}}A_{0}^{1}} + {H_{0}D_{0}^{1}}} \right\rbrack}},} & (167)\end{matrix}$with Det =H₀G₁−G₀H₁ the determinant of the analysis polyphase matrix H_(p) that is expressed in (19). By replacing (166), (167) in equations(164) and (165) the subbands A₁ ¹,D₁ ¹ of the first decomposition levelare expressed as:A ₁ ¹ =Det ⁻¹·(H ₁ G ₁ −zH ₀ G ₀)A ₀ ¹ +Det ⁻¹·(zH ₀ H ₀ −H ₁ H ₁)D ₀¹,  (168)D ₁ ¹ =Det ⁻¹·(G ₁ G ₁ −zG ₀ G ₀)A ₀ ¹ +Det ⁻¹·(zH ₀ G ₀ −H ₁ G ₁)D ₀¹.  (169)In equations (168), (169) the four filters reported in (17), (18) areshown. In addition equations (168), (169) are found as a special case of(3) for k=1,x=1. Hence they consist the proposition P(1).For proposition P(2) (equation (2)) the calculated subbands of level 2are shown pictorially in FIG. 2. Starting from subbands A₁ ²,D₁ ², ifsubband A₀ ₁ is assumed as the input signal, then subbands A₀ ²,D₀ ² andA₁ ²,D₁ ² can be considered as an one-level overcomplete decompositionof A₀ ¹ (a “fictitious” pyramid). For this “fictitious” pyramid,proposition P(1) is applicable, hence:A ₁ ² =F ₀ ¹ A ₀ ² +F ₁ ¹ D ₀ ²,  (170)D ₁ ² =F ₂ ¹ A ₀ ² +F ₃ ¹ D ₀ ².  (171)We perform an inverse wavelet transform in order to calculate A₀ ¹ fromsubbands A₀ ²,D₀ ². The result can be seen in equation (26) with k=1.Now that both subbands A₀ ¹ and D₀ ¹ are known, we can apply propositionP(1) to calculate subband A₁ ¹. The result is equation (27) withk=1,x=1. In this case, by the definitions of (8), the “tail” T(k−1,z) isvoid. With these settings for k,x,T, the procedure to calculate subbandsA₂ ²,D₂ ² is described in equations (29)-(60). Furthermore, theprocedure to calculate subbands A₃ ²,D₃ ² follows equations (67)-(74)with the same settings for k,x,T.

APPENDIX II

First some properties of the polyphase components of H and G are shown.Based on these properties, the roof of equations (93)-(98) isstraightforward. As shown in the example of Table I, for biorthogonalpoint-symmetric filters with Det⁻¹=−1, G is a linear-phase and H becomeslinear phase under a unit translation. These facts are immediatelyverified fro the perfect reconstruction condition and their mathematicalformulation is:G(z)=G(z ⁻¹)

G(z ^(1/2))=G(z ^(−1/2)),   (172)G(−z)=G(−z ⁻¹)

G(z ^(−1/2))=G(−z ^(−1/2)),   (173)H(z ⁻¹)=z ² H(z)

H(z ^(−1/2))=z·H(z ^(1/2)),   (174)H(−z ⁻¹)=z ² H(−z)

H(−z ^(−1/2))=z·H(−z ^(1/2)).   (175)

The Type I polyphase components of H are:H ₀(z)=½(H(z ^(1/2))+H(−z ^(1/2))),   (176)H ₁(z)=½z ^(1/2)(H(z ^(1/2))−H(−z ^(1/2))),   (177)By substituting z with z⁻¹ in equation (176) and utilizing equations(174), (175) for components H(z^(−1/2)) and H(−z^(−1/2)), we get:H ₀(z ⁻¹)=z·H ₀(z).   (178)

Similarly, by substituting z with z⁻¹ in equation (177) and utilizingequations (174), (175) for components H(z^(−1/2)) and H(−z^(−1/2)), weget:H ₁(z ⁻¹)=H₁(z).   (179)

The identical procedure for the Type I polyphase components of G, withthe utilization of equations (172), (173) derives the followingrelations:G ₀(z ⁻¹)=G ₀(z),   (180)G ₁(z ⁻¹)=z ⁻¹ G ₁(z),   (181)

The proof of the equations (93)-(98) is concluded by replacing z withz⁻¹ in the form of each filter F₀ ¹(z),F₁ ¹(z),F₂ ¹(z),F₃ ¹(z) (which isseen in (17), (18)) and replacing the resulting componentsG₀(z⁻¹),G₁(z⁻¹), H₀(z⁻¹),H₁(z⁻¹) by equations (178)-(181).

APPENDIX III

From equation (93), the following properties are immediately derived:F ₀ ¹(z ^(−1/2))=z ^(−1/2) F ₀ ¹(z ^(1/2)),   (182)F ₀ ¹(−z ^(−1/2))=−z ^(−1/2) F ₀ ¹(−z ^(1/2)),   (183)

Using equations (182), (183) and the Type I polyphase components of F₀¹(z) with the replacement of z with z⁻¹, we derive:F _(0,0) ¹(z ⁻¹)=½(F ₀ ¹(z ^(−1/2))+F ₀ ¹(−z ^(−1/2)))

F _(0,0) ¹(z ⁻¹)=½z ^(−1/2)(F ₀ ¹(z ^(1/2))−F ₀ ¹(−z ^(1/2)))=z⁻¹ F_(0,1) ¹(z),  (184)F _(0,1) ¹(z ⁻¹)=½z ^(−1/2)(F ₀ ¹(z ^(−1/2))−F ₀ ¹(−z ^(−1/2)))

F _(0,1) ¹(z ⁻¹)=½z ⁻¹(F ₀ ¹(z ^(1/2))+F ₀ ¹(−z ^(1/2))=z ⁻¹ F _(0,0)¹(z).   (185)

the proof of equations (99)-(102) is concluded by replacing filters F₄²(z⁻¹),F₅ ²(z⁻¹), F₆ ²(z⁻¹), F₇ ²(z⁻¹) from equations (13)-(16)respectively (with k=2 and with the replacement of z with z⁻¹) and thereplacement of F_(0,0) ¹(z⁻¹), F_(0,1) ¹(z⁻¹) and F₀ ¹(z⁻¹), F₁ ¹(z⁻¹),F₂ ¹(z⁻¹), F₃ ¹(z⁻¹) from equations (184), (185) and (93)-(96)respectively Section II contains the preliminary discussion about theLBS method and the introduction to the prediction filter approach.Section III present the proof of the generic formulation of theprediction filters and the proof of their symmetry properties, allowingefficient implementation. Section IV presents the complexity analysis ofthe prediction-filters algorithm and a comparison with the complexity ofthe LBS method when both methods operate in a level-by-level manner andare implemented with convolution. The analysis is formalized in theone-dimensional case for the sake of simplicity in the description, butthe two-dimensional application of the methods is described as well.Finally section V discusses the presented results.

REFERENCES

-   [1] G. Karlsson and M. Vetterli, “Three dimensional subband coding    of video,” in Proc. ICASSP'88, vol. 3, pp. 1100-1103.-   [2] A. S. Lewis and G. Knowles, “video compression using 3D wavelet    transforms,” Electronics Letters, vol. 26, no. 6, pp. 396-398, Mar.    1990.-   [3] J.-R. Ohm: “Three-dimensional subband coding with motion    compensation,” IEEE Trans. Image Processing, vol. 3, no. 5, pp.    559-571, September 1994.-   [4] D. Taubman and A. Zakhor, “Multirate 3-D subband coding of    video,” IEEE Trans. Image Proc., vol. 3, pp. 572-588, September    1994.-   [5] B.-J. Kim, Z. Xiong and W. A. Pearlman, “Low bit-rate scalable    video coding with 3-D Set Partitioning in Hierarchical Trees (3-D    SPIHT),” IEEE Trans. Circuits Syst. Video Technol., vol. 10, no. 8,    pp. 1374-1387, December 2000.-   [6] V. Bottreau, M. Benetiere M, B. Felts, B. Pesquet-Popescu, “A    fully scalable 3D subband video codec,” in Proc. ICIP'01, vol. 2,    pp. 1017-1020.-   [7] A. Said and W. A. Pearlman, “A new fast and efficient image    codec based on Set Partitioning in Hierarchical Trees,” IEEE Trans.    Circuits Syst. Video Technol., vol. 6, no. 3, pp. 243-250, June    1996.-   [8] A. Munteanu, J. Cornelis, G. Van der Auwera and P. Cristea,    “Wavelet-based lossless compression scheme with progressive    transmission capability,” Int. J. Imaging Syst. Technol., John Wiley    & Sons, vol. 10, no. 1, pp. 76-85, January 1999.-   [9] D. Taubman, “High Performance Scalable Image Compression with    EBCOT,” IEEE Trans. Image Proc., vol. 9, no 7, pp. 1158-1170, July    2000.-   [10] P. J. Tourtier, M. Pécot and J.-F. Vial, “Motion compensated    subband coding schemes for compatible High Definition TV coding,”    Signal Proc.: Image Commmun., vol. 4, pp. 325-344, 1992.-   [11] S. J. Choi and J. W. Woods “Motion-compensated 3-D subband    coding of Video”, IEEE Trans. Image Proc., vol. 8, pp. 155-167,    February 1999.-   [12] P. J. Tourtier, M. Pecot and J. F. Vial, “Motion-compensated    subband coding schemes for compatible high definition TV coding”,    Signal Proc.: Image Commmun., vol. 4, pp. 325-344, 1992.-   [13] G. Van der Auwera. A. Munteanu, G. Lafruit, and J. Cornelis,    “Video Coding Based on Motion Estimation in the Wavelet Detail    Images,” in Proc. ICASSP-98, pp. 2801-2804.-   [14] H. W. Park and H. S. Kim, “Motion estimation using    Low-Band-Shift method for wavelet-based moving-picture coding,” IEEE    Trans. Image Proc., vol. 9, no. 4, pp. 577-587, April 2000.-   [15] K. Sivaramakrishnan and T. Nguyen, “A uniform transform domain    video codec based on Dual Tree Complex Wavelet Transform,” in Proc.    ICCASSP'01, vol. 3, pp. 1821-1824.-   [16] H. Sari-Sarraf and D. Brzakovic, “A Shift-Invariant Discrete    Wavelet Transform,” IEEE Trans. Signal Proc., vol. 45, no. 10, pp.    2621-2626, October 1997.-   [17] G. Strang and T. Nguyen, Wavelets and Filer Banks.    Wellesley-Cambridge Press, 1996.-   [18] N. G. Kingsbury, “Shift invariant properties of Dual-Tree    Complex Wavelet Transform ”, in Proc. ICASP'99, vol. 3, pp.    1221-1224.-   [19] I. W. Selesnick, “Hilbert transform pairs of wavelet bases,”    Signal Proc. Letters, vol. 8, no. 6, pp 170-173, June 2001.-   [20] G. Strang and T. Nguyen, Wavelets and Filer Banks.    Wellesley-Cambridge Press, 1996.-   [21] J. Kova{hacek over (c)}ević and Martin Vetterli, “Nonseparable    Multidimensional Perfect Reconstruction Filter Banks and Wavelet    Bases for R^(n),” IEEE Trans. Inform. Theory, vol. 38, pp 553-555,    no. March 1992.

1. A method of digital encoding or decoding a video bit stream, the bitstream comprising a representation of a sequence of n-dimensional datastructures, the method comprising: providing a set of one or moresubsampled subbands forming a multilevel subband transform of one datastructure of the sequence; and after providing the set, inputting atleast a part of the set to at least one digital filter so as to generatea further set of one or more subbands of a shifted version of said datastructure, wherein said shifted version of said data structure is atleast one of temporally shifted and spatially shifted in a video frame,and wherein the further set is generated based on the provided set, andthe further set has at least one subsampled subband that is not includedin the one or more subsampled subbands of the provided set.
 2. Themethod according to claim 1, further comprising mapping at least a partof the data of one data structure of the sequence within predefinedsimilarity criteria to at least a part of the data of another datastructure of the sequence.
 3. The method according to claim 1, whereinapplying at least one digital filter includes applying the digitalfilter only to members of the set of subsampled subbands of thetransform of the data structure.
 4. The method according to claim 1,wherein the digital filter is characterized by at least two non-zerovalues.
 5. The method according to claim 4, wherein the part of the dataof one data structure comprises one block.
 6. The method according toclaim 1, wherein the digital subband transform comprises a wavelet. 7.The method according to claim 1, wherein the data structures comprisedata frames and the set of subsampled subbands of the transform of thedata structure define a reference frame.
 8. The method according toclaim 1, wherein the generation of the further set of one or morefurther subbands is performed in a level-by-level manner, wherein eachlevel is a level of a subband pyramid containing the subbands of thesubband transform.
 9. The method of claim 1, wherein the digital filtergenerates the further set of one or more subbands at a single rate. 10.The method according to claim 1, further comprising transforming a videobit stream into the multilevel subband transform.
 11. The methodaccording to claim 1, further comprising, prior to inputting the atleast part of the set to the at least one digital filter, selecting aset of one or more subsampled subbands of a shifted version of said datastructure to be generated.
 12. The method according to claim 11, furthercomprising prior to inputting the at least part of the set to the atleast one digital filter, selecting an unneeded set of one or moresubsampled subbands of a shifted version of said data structure to notbe generated.
 13. A computer readable medium comprising executablemachine readable code which, when executed performs digital filtering toat least a part of a previously received set of subsampled subbandsforming a multilevel subband transform of a video data structure togenerate a further set of one or more subbands of a shifted version ofsaid data structure, wherein said shifted version of said data structureis at least one of temporally shifted and spatially shifted in a videoframe, and wherein the further set is generated based on the previouslyreceived set, and the further set has at least one subsampled subbandthat is not included the one or more subsampled subbands of the receivedset.
 14. The method of claim 13, wherein the digital filtering generatesthe further set of one or more subbands at a single rate.
 15. A methodof digital encoding or decoding a video bit stream, the bit streamcomprising a representation of a sequence of n-dimensional datastructures, the method comprising: receiving one or more subsampledsubbands of an overcomplete representation of the data structure, theovercomplete representation comprising a shifted version of the datastructure, wherein said shifted version of said data structure is atleast one of temporally shifted and spatially shifted in a video frame,and the subsampled subbands forming a multilevel subband transform of atleast one data structure of the sequence; and after receiving thesubbands, applying at least one digital filter to at least one of theone or more subsampled subbands to generate an additional one or moresubbands of the overcomplete representation of the data structure,wherein the digital filter generates the additional one or more subbandsat a single rate, and the additional one or more subbands are at thesame level as the received one or more subsampled subbands, and whereinthe additional subbands are generated based on the received subbands,and the further set has at least one subsampled subband that is notincluded in the one or more subsampled subbands of the received set. 16.The method according to claim 15, wherein the generation of theadditional one or more subbands is performed according to alevel-by-level method.
 17. An apparatus for encoding or decoding adigital bit stream, the bit stream comprising a representation of asequence of n-dimensional video data structures, the method comprising:means for receiving a set of one or more subsampled subbands forming amultilevel subband transform of one data structure of the sequence; andmeans for after receiving the set, generating a further set of one ormore subbands of a shifted version of said data structure based at leastin part on at least a part of the set of one or more subsampledsubbands, wherein said shifted version of said data structure is atleast one of temporally shifted and spatially shifted in a video frame,and wherein the further set is generated based on the received set, andthe further set has at least one subsampled subband that is not includedin the one or more subsampled subbands of the received set.
 18. Themethod according to claim 17, wherein the means for generating isconfigured to generate the further set of one or more subbands accordingto a level-by-level method.
 19. The method according to claim 17,wherein the means for generating is configured to generate the furtherset of one or more subbands at the same level as the received one ormore subsampled subbands.
 20. A method of digital encoding or decoding avideo bit stream, the bit stream comprising a representation of asequence of n-dimensional data structures, the method comprising:transforming a video bit stream into a multilevel subband transform ofone data structure of the sequence; and after the transforming, applyingat least one digital filter to the set of subsampled subbands of thedata structure to generate a further set of one or more subbands of ashifted version of said data structure, wherein said shifted version ofsaid data structure is at least one of temporally shifted and spatiallyshifted in a video frame, and the further set has at least onesubsampled subband that is not included in the one or more subsampledsubbands of the video stream.
 21. A method of digital encoding ordecoding a video bit stream, the bit stream comprising a representationof a sequence of n-dimensional data structures, the method comprising:providing a set of one or more subsampled subbands forming a multilevelsubband transform of one data structure of the sequence; after providingthe set, inputting a part of the set to at least one digital filter soas to generate a further set of one or more subbands of a shiftedversion of said data structure, wherein said shifted version of saiddata structure is at least one of temporally shifted and spatiallyshifted in a video frame, and wherein the further set is generated basedon the provided set, and the further set has at least one subsampledsubband that is not included in the one or more subsampled subbands ofthe provided set; and providing the further set to a motion compensationmodule configured to perform motion compensation; and generating amotion compensated video frame with the further set of subbands.